13750:
11992:
14762:
14242:
11952:
11972:
11962:
11982:
1182:
5155:
1020:
4952:
2129:
2892:
1177:{\displaystyle \left({\frac {a}{p}}\right)={\begin{cases}\;\;\,0&{\text{ if }}p{\text{ divides }}a\\+1&{\text{ if }}a\operatorname {R} p{\text{ and }}p{\text{ does not divide }}a\\-1&{\text{ if }}a\operatorname {N} p{\text{ and }}p{\text{ does not divide }}a\end{cases}}}
3996:
4358:
5653:), then follow the algorithm described in congruence of squares. The efficiency of the factoring algorithm depends on the exact characteristics of the root-finder (e.g. does it return all roots? just the smallest one? a random one?), but it will be efficient.
2310:
4237:
1866:
3231:
3109:
3857:
5150:{\displaystyle x\equiv {\begin{cases}\pm \;a^{(n+3)/8}{\pmod {n}}&{\text{ if }}a{\text{ is a quartic residue modulo }}n\\\pm \;a^{(n+3)/8}2^{(n-1)/4}{\pmod {n}}&{\text{ if }}a{\text{ is a quartic non-residue modulo }}n\end{cases}}}
4126:
3613:
1623:≡ 1 (mod 840), and Dirichlet's theorem says there are an infinite number of primes of this form. 2521 is the smallest, and indeed 1 ≡ 1, 1046 ≡ 2, 123 ≡ 3, 2 ≡ 4, 643 ≡ 5, 87 ≡ 6, 668 ≡ 7, 429 ≡ 8, 3 ≡ 9, and 529 ≡ 10 (mod 2521).
1993:
1748:
2718:
3868:
4933:
3751:
3676:
5379:
5594:
5483:
872:
The fact that, e.g., modulo 15 the product of the nonresidues 3 and 5, or of the nonresidue 5 and the residue 9, or the two residues 9 and 10 are all zero comes from working in the full ring
5319:
4252:
5820:
5533:
5426:
4846:
911:
859:
405:
11148:
5217:
4793:
2348:
2204:
676:
Modulo 8, the product of the nonresidues 3 and 5 is the nonresidue 7, and likewise for permutations of 3, 5 and 7. In fact, the multiplicative group of the non-residues and 1 form the
109:
4141:
182:(1801). Article 95 introduces the terminology "quadratic residue" and "quadratic nonresidue", and states that if the context makes it clear, the adjective "quadratic" may be dropped.
2957:
1763:
1327:
1534:
1493:
1391:
1250:
1978:
1432:
3467:
172:, and other number theorists of the 17th and 18th centuries established theorems and formed conjectures about quadratic residues, but the first systematic treatment is § IV of
3124:
3005:
3762:
300:
277:
4018:
5649:, and have the efficient square root algorithm find a root. Repeat until it returns a number not equal to the one we originally squared (or its negative modulo
1191:) are treated specially. As we have seen, it makes many formulas and theorems easier to state. The other (related) reason is that the quadratic character is a
3511:
2146:
An intriguing fact about these two theorems is that all known proofs rely on analysis; no-one has ever published a simple or direct proof of either statement.
1263:
One advantage of this notation over Gauss's is that the
Legendre symbol is a function that can be used in formulas. It can also easily be generalized to
869:, and there is no simple rule that predicts which one their product will be in. Modulo a prime, there is only the subgroup of squares and a single coset.
12129:
2124:{\displaystyle L(1)={\frac {\pi }{\left(2-\left({\frac {2}{q}}\right)\right)\!{\sqrt {q}}}}\sum _{n=1}^{\frac {q-1}{2}}\left({\frac {n}{q}}\right)>0.}
747:
Modulo a composite number, the product of two residues is a residue. The product of a residue and a nonresidue may be a residue, a nonresidue, or zero.
1576:
770:
The product of the residue 3 and the nonresidue 5 is the residue 3, whereas the product of the residue 4 and the nonresidue 2 is the nonresidue 2.
1666:
12804:
11373:. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the
5612:
allows us to find the roots efficiently. Say there were an efficient algorithm for finding square roots modulo a composite number. The article
2887:{\displaystyle A_{ij}=\left\{k\in \{1,2,\dots ,p-2\}:\left({\frac {k}{p}}\right)=(-1)^{i}\land \left({\frac {k+1}{p}}\right)=(-1)^{j}\right\},}
809:
This phenomenon can best be described using the vocabulary of abstract algebra. The congruence classes relatively prime to the modulus are a
665:
is a nonresidue, and in general all the residues and nonresidues obey the same rules, except that the products will be zero if the power of
12887:
12028:
3991:{\displaystyle \left|\sum _{n=M+1}^{M+N}\left({\frac {n}{q}}\right)\right|<{\frac {4}{\pi ^{2}}}{\sqrt {q}}\log q+0.41{\sqrt {q}}+0.61.}
8201:, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 19, 23, 24, 25, 27, 32, 35, 36, 37, 38, 41, 46, 48, 49, 50, 54, 55, 57, 61, 64, 65, 67, 69, 70, 71, 72
11975:
8070:, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 19, 20, 24, 25, 27, 29, 30, 32, 36, 37, 38, 40, 43, 45, 48, 49, 50, 54, 57, 58, 60, 64
6248:
6238:
6221:
5935:
4538:
4492:
1196:
420:
Modulo a prime, the product of two nonresidues is a residue and the product of a nonresidue and a (nonzero) residue is a nonresidue.
360:
228:) to some in the obtained list. But the obtained list is not composed of mutually incongruent quadratic residues (mod n) only. Since
10930:, pp. 135–137, (proof of P–V, (in fact big-O can be replaced by 2); journal references for Paley, Montgomery, and Schur)
414:
Following this convention, the multiplicative inverse of a residue is a residue, and the inverse of a nonresidue is a nonresidue.
13201:
11675:
7672:, 1, 4, 6, 9, 10, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 29, 33, 35, 36, 37, 39, 40, 47, 49, 54, 55, 56, 59, 60, 62, 64, 65
1615:≡ 1 (mod 8), (mod 12), (mod 5) and (mod 28), then by the law of quadratic reciprocity 2, 3, 5, and 7 will all be residues modulo
4005:
805:
The product of the nonresidues 2 and 8 is the residue 1, whereas the product of the nonresidues 2 and 7 is the nonresidue 14.
13786:
13359:
11542:
10989:
12147:
10952:
Pomerance & Crandall, ex 2.38 pp.106–108. result from T. Cochrane, "On a trigonometric inequality of
Vinogradov",
14227:
13214:
12537:
4862:
3695:
3628:
6085:
5959:
5172:
found efficient algorithms that work for all prime moduli. Both algorithms require finding a quadratic nonresidue modulo
4353:{\displaystyle \max _{N}\left|\sum _{n=1}^{N}\left({\frac {d}{n}}\right)\right|>{\frac {1}{7}}{\sqrt {d}}\log \log d}
14029:
13966:
13219:
13209:
12946:
12799:
12152:
6206:
which will yield a factorization. The number field sieve is the fastest general-purpose factorization algorithm known.
5328:
12143:
5546:
5435:
14278:
13355:
11578:
11560:
11508:
11479:
11461:
11443:
11421:
11399:
11065:
11029:
6019:
5698:
5278:
12697:
7121:, 1, 3, 4, 5, 9, 12, 13, 14, 15, 16, 19, 20, 22, 25, 27, 34, 36, 39, 41, 42, 45, 46, 47, 48, 49, 52, 56, 57, 58, 60
6151:
is based on the same principles. There is a deterministic version of it, but the proof that it works depends on the
5782:
5495:
5388:
2305:{\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}\cdot {\frac {q-1}{2}}}}
13452:
13196:
12021:
11391:
10910:
Crandall & Pomerance, ex 2.38, pp 106–108 discuss the similarities and differences. For example, tossing
10815:
10633:
6191:
5176:, and there is no efficient deterministic algorithm known for doing that. But since half the numbers between 1 and
4810:
875:
823:
369:
12757:
12450:
11118:
11013:
6148:
5187:
4763:
4232:{\displaystyle \max _{N}\left|\sum _{n=1}^{N}\left({\frac {n}{q}}\right)\right|>{\frac {1}{2\pi }}{\sqrt {q}}}
4001:
2318:
12191:
2142:
For example, modulo 11 there are four residues less than 6 (namely 1, 3, 4, and 5), but only one nonresidue (2).
13713:
13415:
13178:
13173:
12998:
12419:
12103:
11706:
6981:, 1, 3, 4, 5, 7, 9, 12, 15, 16, 17, 19, 20, 21, 22, 25, 26, 27, 28, 29, 35, 36, 41, 45, 46, 48, 49, 51, 53, 57
6152:
6039:
4545:
congruent to 1 mod 4, the excess is zero, since −1 is a quadratic residue and the residues are symmetric under
4433:
4009:
1861:{\displaystyle L(1)=-{\frac {\pi }{\sqrt {q}}}\sum _{n=1}^{q-1}{\frac {n}{q}}\left({\frac {n}{q}}\right)>0.}
1633:
62:
14796:
14716:
13708:
13491:
13408:
13121:
13052:
12929:
12171:
11021:
10981:
6035:
5694:
417:
Following this convention, modulo an odd prime number there are an equal number of residues and nonresidues.
2903:
955:
Although it makes things tidier, this article does not insist that residues must be coprime to the modulus.
14564:
13961:
13945:
13633:
13459:
13145:
12779:
12378:
11965:
11752:
10648:
6180:
5220:
5165:
4796:
1641:
1584:
1285:
424:
1498:
1457:
1352:
1209:
14791:
13981:
13511:
13506:
13116:
12855:
12784:
12113:
12014:
11747:
11732:
11668:
11357:
6104:) using a modification of Euclid's algorithm, and also using Euler's criterion. If the results disagree,
1939:
1875:≡ 3 (mod 4)), the sum of the quadratic residues minus the sum of the nonresidues in the range 1, 2, ...,
1396:
259:) around its midpoint, hence it is actually only needed to square all the numbers in the list 0, 1, ...,
178:
3226:{\displaystyle \alpha _{01}={\frac {p+1}{4}},\;\alpha _{00}=\alpha _{10}=\alpha _{11}={\frac {p-3}{4}}.}
1932:≡ 1 (mod 4), the sum of the quadratic residues minus the sum of the nonresidues in the range 1, 2, ...,
14786:
14737:
14495:
13826:
13440:
13030:
12424:
12392:
12083:
6199:
3482:
3437:
3104:{\displaystyle \alpha _{00}={\frac {p-5}{4}},\;\alpha _{01}=\alpha _{10}=\alpha _{11}={\frac {p-1}{4}}}
13779:
13730:
13679:
13576:
13074:
13035:
12512:
12157:
11927:
11886:
11765:
5717:
4698:
1588:
12186:
6488:, 1, 4, 6, 7, 9, 10, 11, 13, 15, 16, 17, 24, 25, 28, 29, 36, 37, 38, 40, 42, 43, 44, 46, 47, 49, 52
4967:
1050:
14742:
14732:
14441:
14061:
13971:
13571:
13501:
13040:
12892:
12875:
12598:
12078:
11771:
11245:
10852:
6167:, then the GRH would be false, which would have implications through many branches of mathematics.
2134:
This implies that there are more quadratic residues than nonresidues among the numbers 1, 2, ..., (
1000:
995:
Although this notation is compact and convenient for some purposes, a more useful notation is the
14747:
14451:
14341:
14222:
13910:
13403:
13380:
13341:
13227:
13168:
12814:
12734:
12578:
12522:
12135:
11991:
11714:
11698:
11534:
11378:
6202:) generate small quadratic residues (modulo the number being factorized) in an attempt to find a
5963:
5219:
until a nonresidue is found will quickly produce one. A slight variant of this algorithm is the
3852:{\displaystyle \left|\sum _{n=M+1}^{M+N}\left({\frac {n}{q}}\right)\right|<{\sqrt {q}}\log q.}
14629:
14554:
14475:
14465:
14431:
14381:
14123:
14022:
13940:
13935:
13930:
13808:
13693:
13420:
13398:
13365:
13258:
13104:
13089:
13062:
13013:
12897:
12832:
12657:
12623:
12618:
12492:
12323:
12300:
11955:
11775:
11724:
11661:
10653:
5169:
1580:
138:
11995:
4121:{\displaystyle \left|\sum _{n=M+1}^{M+N}\chi (n)\right|=O\left({\sqrt {q}}\log \log q\right).}
285:
262:
14624:
14618:
14391:
14271:
14194:
13920:
13900:
13623:
13476:
13268:
12986:
12722:
12628:
12487:
12472:
12353:
12328:
11932:
11861:
6203:
6000:
5613:
5604:(i.e. an efficient solution to either problem could be used to solve the other efficiently).
5597:
4939:
4853:
4800:
2155:
169:
165:
146:
4693:
The theoretical way solutions modulo the prime powers are combined to make solutions modulo
14588:
14511:
14469:
14412:
14334:
14054:
13986:
13905:
13772:
13596:
13558:
13435:
13239:
13079:
13003:
12981:
12809:
12767:
12666:
12633:
12497:
12285:
12196:
11922:
11757:
10638:
10628:
6187:
6057:
4804:
1657:
1637:
1253:
356:
11985:
11075:
11039:
10999:
8:
14645:
14569:
14515:
14501:
14491:
14435:
14376:
14361:
14351:
14163:
14158:
14143:
14138:
14049:
13799:
13725:
13616:
13601:
13581:
13538:
13425:
13375:
13301:
13246:
13183:
12976:
12971:
12919:
12687:
12676:
12348:
12248:
12176:
12167:
12163:
12098:
12093:
11891:
11800:
11795:
11789:
11781:
11742:
11292:
4608:
An important difference between prime and composite moduli shows up here. Modulo a prime
3608:{\displaystyle \left|\sum _{n=M+1}^{M+N}\chi (n)\right|=O\left({\sqrt {q}}\log q\right),}
3486:
810:
364:
41:
11981:
14604:
14521:
14425:
14356:
14324:
14303:
14298:
14189:
14148:
14071:
13976:
13843:
13838:
13764:
13754:
13523:
13486:
13471:
13464:
13447:
13251:
13233:
13099:
13025:
13008:
12961:
12774:
12683:
12517:
12502:
12462:
12414:
12399:
12387:
12343:
12318:
12088:
12037:
11937:
11881:
11785:
11702:
11497:
11274:
6046:
6031:
6027:
818:
134:
12707:
11629:
529:
So a nonzero number is a residue mod 8, 16, etc., if and only if it is of the form 4(8
14695:
14665:
14535:
14445:
14396:
14329:
14245:
14133:
14091:
14081:
14015:
13821:
13749:
13689:
13496:
13306:
13296:
13188:
13069:
12904:
12880:
12661:
12645:
12550:
12527:
12404:
12373:
12338:
12233:
12068:
11851:
11626:
11609:
11574:
11556:
11538:
11527:
11504:
11475:
11457:
11439:
11417:
11395:
11061:
11025:
10985:
8113:, 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 17, 18, 21, 24, 25, 27, 28, 32, 34, 36, 37, 42
5989:
5250:
4387:) is more subtle, but it is always prime, with 7 appearing for the first time at 71.
1264:
774:
Also, the product of two nonresidues may be either a residue, a nonresidue, or zero.
359:. In this case, it is customary to consider 0 as a special case and work within the
14765:
14650:
14319:
14264:
14096:
13874:
13867:
13862:
13703:
13698:
13591:
13548:
13370:
13331:
13326:
13311:
13137:
13094:
12991:
12789:
12739:
12313:
12275:
11856:
11841:
11643:
11604:
11492:
11488:
11366:
11266:
11071:
11035:
10995:
10969:
5986:
5982:
5273:
929:
677:
34:
13996:
11018:
Ten
Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis
1915:
1 + 4 + 9 + 5 + 3 = 22, 2 + 6 + 7 + 8 + 10 = 33, and the difference is −11.
14685:
14675:
14594:
14421:
14346:
14215:
14153:
14086:
13884:
13879:
13831:
13816:
13684:
13674:
13628:
13611:
13566:
13528:
13430:
13350:
13157:
13084:
13057:
13045:
12951:
12865:
12839:
12794:
12762:
12563:
12365:
12308:
12258:
12223:
12181:
11870:
11846:
11761:
11588:
11370:
11362:
11053:
10943:. The proof is a page long and only requires elementary facts about Gaussian sums
10918:/2 heads followed by that many tails. V-P inequality rules that out for residues.
7707:, 1, 4, 6, 9, 10, 11, 13, 14, 15, 16, 17, 21, 23, 24, 25, 31, 35, 36, 38, 40, 41
6195:
5777:
4758:
2351:
1619:, and thus all numbers 1–10 will be. The CRT says that this is the same as
1203:
996:
8236:, 1, 2, 4, 8, 9, 11, 15, 16, 18, 22, 23, 25, 29, 30, 32, 36, 37, 39, 43, 44, 46
6163:
is prime or the GRH is false". If the second output ever occurs for a composite
4379:
is clearly 1. The question of the magnitude of the least quadratic non-residue
3478:
642:
Notice that the rules are different for powers of two and powers of odd primes.
14700:
14689:
14608:
14459:
14371:
14113:
13855:
13850:
13669:
13648:
13606:
13586:
13481:
13336:
12934:
12924:
12914:
12909:
12843:
12717:
12593:
12482:
12477:
12455:
12056:
11907:
11826:
11710:
10973:
6141:
3619:
1645:
814:
1556:
Although quadratic residues appear to occur in a rather random pattern modulo
14780:
14614:
14579:
13643:
13321:
12828:
12613:
12603:
12573:
12558:
12228:
11866:
11718:
11684:
11435:
10643:
6179:
Gauss discusses two factoring algorithms that use quadratic residues and the
4426:
4243:
1275:
45:
20:
1743:{\displaystyle L(s)=\sum _{n=1}^{\infty }\left({\frac {n}{q}}\right)n^{-s}.}
924:
For this reason some authors add to the definition that a quadratic residue
14598:
14531:
14505:
14386:
14179:
13991:
13543:
13390:
13291:
13283:
13163:
13111:
13020:
12956:
12939:
12870:
12729:
12588:
12290:
12073:
11912:
11836:
11736:
11522:
7516:, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40
6023:
1192:
1008:
914:
341:
142:
14669:
14655:
14573:
14559:
14525:
13653:
13533:
12712:
12702:
12649:
12333:
12253:
12238:
12118:
12063:
11518:
11390:, translated by Clarke, Arthur A. (Second corrected ed.), New York:
6014:
The fact that finding a square root of a number modulo a large composite
5974:
5713:
5231:
4738:
x ≡ 7 (mod 3) has two solutions, 1 and 2; x ≡ 7 (mod 5) has no solutions.
4725:
x ≡ 4 (mod 3) has two solutions, 1 and 2; x ≡ 4 (mod 5) has two, 2 and 3.
4132:
1270:
There is a generalization of the
Legendre symbol for composite values of
302:. The list so obtained may still contain mutually congruent numbers (mod
130:
14679:
14659:
14584:
13795:
12583:
12438:
12409:
12215:
11917:
11876:
11728:
11647:
11278:
5904:, but then this takes the problem away from quadratic residues (unless
306:). Thus, the number of mutually noncongruent quadratic residues modulo
11499:
Computers and
Intractability: A Guide to the Theory of NP-Completeness
5265:
has been factored into prime powers the solution was discussed above.
411:
has a multiplicative inverse. This is not true for composite moduli.)
14184:
13925:
13735:
13638:
12691:
12608:
12568:
12532:
12468:
12280:
12270:
12243:
12006:
11634:
11416:], translated by Maser, H. (second ed.), New York: Chelsea,
11374:
3474:
3402:
Gauss (1828) introduced this sort of counting when he proved that if
1257:
11270:
7148:, 1, 3, 4, 7, 9, 10, 11, 12, 16, 21, 25, 26, 27, 28, 30, 33, 34, 36
4659:
is written as a product of powers of distinct primes, and there are
244:), the list obtained by squaring all numbers in the list 1, 2, ...,
14455:
14417:
14366:
14106:
14038:
13915:
13720:
13518:
12966:
12671:
12265:
11293:"The design and application of modular acoustic diffusing elements"
862:
5995:
Paley digraphs are directed analogs of Paley graphs, one for each
5858:
Note: This theorem essentially requires that the factorization of
4712:
x ≡ 6 (mod 3) has one solution, 0; x ≡ 6 (mod 5) has two, 1 and 4.
14287:
14101:
13316:
12108:
5704:
On the other hand, if we want to know if there is a solution for
24:
11653:
11414:
2984:
is the number of nonresidues that are followed by a residue, and
11816:
4728:
and there are four solutions modulo 15, namely 2, 7, 8, and 13.
4451:
2991:
is the number of nonresidues that are followed by a nonresidue.
1572:, their distribution also exhibits some striking regularities.
653:, the products of residues and nonresidues relatively prime to
465:
is a nonresidue and the negative of a nonresidue is a residue.
454:
is a residue and the negative of a nonresidue is a nonresidue.
157:
137:, quadratic residues are now used in applications ranging from
473:
All odd squares are ≡ 1 (mod 8) and thus also ≡ 1 (mod 4). If
193:
may be obtained by simply squaring all the numbers 0, 1, ...,
14210:
12860:
12206:
12051:
6018:
is equivalent to factoring (which is widely believed to be a
866:
407:. (In other words, every congruence class except zero modulo
173:
161:
152:
4807:. If it is −1 there is no solution. Secondly, assuming that
2977:
is the number of residues that are followed by a nonresidue,
14007:
10939:
Planet Math: Proof of Pólya–Vinogradov
Inequality in
8449:
8445:
6252:
6242:
6216:
5930:
5928:
1, 2, 2, 2, 3, 4, 4, 3, 4, 6, 6, 4, 7, 8, 6, ... (sequence
5143:
4533:
4487:
1170:
14256:
11624:
6006:
The construction of these graphs uses quadratic residues.
5608:
The above discussion indicates how knowing the factors of
1197:
multiplicative group of nonzero congruence classes modulo
5689:
by computing the
Legendre symbol. However, for composite
3485:
proved (independently) in 1918 that for any nonprincipal
2970:
is the number of residues that are followed by a residue,
1920:
In fact the difference will always be an odd multiple of
971:
to denote residuosity and non-residuosity, respectively;
13794:
6049:
is a similar problem that is also used in cryptography.
5645:
efficiently. Generate a random number, square it modulo
3681:
this shows that the number of quadratic residues modulo
431:≡ 1 (mod 4) then −1 is a quadratic residue modulo
10876:
Lemmermeyer, p. 29 ex. 1.22; cf pp. 26–27, Ch. 10
10801:
This extension of the domain is necessary for defining
1561:
6683:, 1, 2, 4, 5, 7, 8, 9, 10, 14, 16, 18, 19, 20, 25, 28
5791:
5555:
5504:
5444:
5397:
5337:
5287:
4715:
and there are two solutions modulo 15, namely 6 and 9.
4519:) is the number of quadratic residues on the range (0,
3445:
1506:
1465:
1404:
1360:
1293:
1217:
683:
11831:
11821:
11434:, Algorithmic Number Theory, vol. I, Cambridge:
11331:, pp. 109–110; Euler's criterion requires O(log
11121:
10739:
10737:
5962:
have been based on number-theoretic concepts such as
5949:
5785:
5549:
5498:
5438:
5391:
5331:
5281:
5190:
4955:
4928:{\displaystyle x\equiv \pm \;a^{(n+1)/4}{\pmod {n}},}
4865:
4813:
4766:
4701:; it can be implemented with an efficient algorithm.
4568:
4255:
4144:
4021:
3871:
3765:
3746:{\displaystyle {\frac {1}{2}}N+O({\sqrt {q}}\log q).}
3698:
3671:{\displaystyle \chi (n)=\left({\frac {n}{q}}\right),}
3631:
3514:
3440:
3127:
3008:
2906:
2721:
2321:
2207:
1996:
1942:
1766:
1669:
1551:
1501:
1460:
1399:
1355:
1288:
1212:
1023:
878:
865:
of it. Different nonresidues may belong to different
826:
372:
288:
265:
65:
11595:-Complete Decision Problems for Binary Quadratics",
10709:
10707:
10697:
10695:
6132:
none will. If, after using many different values of
5916:
The list of the number of quadratic residues modulo
2651:
778:For example, from the table for modulus 15
16:
Integer that is a perfect square modulo some integer
10896:
Theorie der biquadratischen Reste, Erste
Abhandlung
10885:
Crandall & Pomerance, ex 2.38, pp 106–108
10839:, pp. 8–9, 43–51. These are classical results.
6508:, 1, 4, 5, 6, 7, 9, 13, 16, 20, 22, 23, 24, 25, 28
6186:Several modern factorization algorithms (including
5911:
5701:as factorization, but is assumed to be quite hard.
751:For example, from the table for modulus 6
11526:
11496:
11208:Crandall & Pomerance, ex. 6.5 & 6.6, p.273
11142:
10968:
10734:
5814:
5588:
5527:
5477:
5420:
5373:
5313:
5211:
5149:
4927:
4840:
4787:
4470:) > X is bounded by a constant depending on ε.
4352:
4231:
4131:This result cannot be substantially improved, for
4120:
3990:
3851:
3745:
3670:
3607:
3461:
3429:
3225:
3103:
2951:
2886:
2342:
2304:
2123:
1972:
1860:
1742:
1528:
1487:
1426:
1385:
1321:
1244:
1176:
905:
853:
399:
294:
271:
133:concept from the branch of number theory known as
103:
11451:
10816:Legendre symbol#Properties of the Legendre symbol
10704:
10692:
6076:is composite the formula may or may not compute (
5374:{\displaystyle \left({\tfrac {a}{n/2}}\right)=-1}
5164:≡ 1 (mod 8), however, there is no known formula.
2052:
1936:− 1 is zero, implying that both sums equal
1548:is prime, the Jacobi and Legendre symbols agree.
439:≡ 3 (mod 4) then −1 is a nonresidue modulo
329:The product of two residues is always a residue.
224:), any other quadratic residue is congruent (mod
14778:
11553:A Classical Introduction to Modern Number Theory
5977:are dense undirected graphs, one for each prime
5942:A formula to count the number of squares modulo
5616:discusses how finding two numbers x and y where
5589:{\displaystyle \left({\tfrac {a}{n/2}}\right)=1}
5478:{\displaystyle \left({\tfrac {a}{n/2}}\right)=1}
4257:
4146:
337:Modulo 2, every integer is a quadratic residue.
11195:, p. 156 ff; the algorithm requires O(log
10914:coins, it is possible (though unlikely) to get
5314:{\displaystyle \left({\tfrac {a}{n}}\right)=-1}
4748:
4485:2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, ... (sequence
2149:
13817:Zero polynomial (degree undefined or −1 or −∞)
11586:
11217:
11179:, p. 156 ff; the algorithm requires O(log
11052:
6224:) lists the quadratic residues mod 1 to 75 (a
6209:
6140:has not been proved composite it is called a "
5815:{\displaystyle \left({\tfrac {a}{p}}\right)=1}
5528:{\displaystyle \left({\tfrac {a}{n}}\right)=1}
5421:{\displaystyle \left({\tfrac {a}{n}}\right)=1}
5184:at random and calculating the Legendre symbol
4390:The Pólya–Vinogradov inequality above gives O(
4370:
2700:, etc., are almost equal. More precisely, let
2616:≡ 1, 5, 7, 9, 19, 25, 35, 37, 39, 43 (mod 44)
14272:
14023:
13780:
12022:
11669:
5981:≡ 1 (mod 4), that form an infinite family of
4841:{\displaystyle \left({\frac {a}{n}}\right)=1}
461:≡ 3 (mod 4) the negative of a residue modulo
450:≡ 1 (mod 4) the negative of a residue modulo
11550:
11487:
10762:
5661:is a quadratic residue or nonresidue modulo
4008:improved this in 1977, showing that, if the
2779:
2749:
906:{\displaystyle (\mathbb {Z} /n\mathbb {Z} )}
854:{\displaystyle (\mathbb {Z} /n\mathbb {Z} )}
400:{\displaystyle (\mathbb {Z} /p\mathbb {Z} )}
289:
266:
11568:
11517:
11452:Crandall, Richard; Pomerance, Carl (2001),
11429:
11328:
11316:
11192:
11176:
11143:{\displaystyle \left({\frac {a}{n}}\right)}
11112:
11088:
6112:may be composite or prime. For a composite
5596:, the problem is known to be equivalent to
5212:{\displaystyle \left({\frac {x}{n}}\right)}
5134: is a quartic non-residue modulo
4788:{\displaystyle \left({\frac {a}{n}}\right)}
2343:{\displaystyle \left({\frac {p}{q}}\right)}
1632:The first of these regularities stems from
1278:, but its properties are not as simple: if
1187:There are two reasons why numbers ≡ 0 (mod
928:must not only be a square but must also be
14279:
14265:
14030:
14016:
13787:
13773:
12214:
12029:
12015:
11971:
11961:
11676:
11662:
11644:Proof of Pólya–Vinogradov inequality
11571:Reciprocity Laws: from Euler to Eisenstein
11454:Prime Numbers: A Computational Perspective
11012:
6246:, and for nonzero quadratic residues, see
6232:). (For the quadratic residues coprime to
5045:
4973:
4875:
3162:
3043:
1054:
1053:
497:For example, mod (32) the odd squares are
189:, a list of the quadratic residues modulo
153:History, conventions, and elementary facts
11608:
11551:Ireland, Kenneth; Rosen, Michael (1990),
11469:
10927:
10864:
10848:
10836:
4604:assuming one does exist, to calculate it?
1055:
896:
883:
844:
831:
521:2 ≡ 6≡ 10 ≡ 14≡ 4
481:= 8, 16, or some higher power of 2, then
390:
377:
104:{\displaystyle x^{2}\equiv q{\pmod {n}}.}
11529:An Introduction to the Theory of Numbers
10964:
10962:
6170:
5877:, then the congruence can be reduced to
5731:is a quadratic residue modulo composite
361:multiplicative group of nonzero elements
11597:Journal of Computer and System Sciences
11555:(second ed.), New York: Springer,
5272:is not congruent to 2 modulo 4 and the
5032: is a quartic residue modulo
2605:≡ 1, 7, 9, 11, 13, 19, 23, 37 (mod 40)
2596:≡ 1, 3, 9, 13, 27, 31, 37, 39 (mod 40)
1627:
14779:
12036:
11474:(third ed.), New York: Springer,
11243:
11228:
10900:Untersuchungen über hohere Arithmetik)
8164:, 1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18
6060:is a formula for the Legendre symbol (
4856:found that the solutions are given by
4425:, obtained by estimates of Burgess on
2952:{\displaystyle \alpha _{ij}=|A_{ij}|.}
1560:, and this has been exploited in such
548:if and only if it is a residue modulo
468:
14260:
14011:
13768:
12010:
11657:
11625:
11430:Bach, Eric; Shallit, Jeffrey (1996),
11410:Untersuchungen über hohere Arithmetik
11407:
11385:
11233:. New York: McGraw HIll. p. 195.
11199:) steps and is also nondeterministic.
10959:
8332:, 1, 4, 6, 9, 11, 14, 16, 19, 21, 24
6092:is prime or composite picks a random
5735:, one can use the following theorem:
5381:, then there is also no solution. If
4741:and there are no solutions modulo 15.
4473:The least quadratic non-residues mod
3430:The Pólya–Vinogradov inequality
1983:Dirichlet also proved that for prime
1591:(CRT) it is easy to see that for any
1322:{\displaystyle ({\tfrac {a}{m}})=-1,}
1267:, quartic and higher power residues.
1256:to be extended to the multiplicative
14228:List of differential geometry topics
11091:, p. 104 ff; it requires O(log
6159:is definitely composite" or "either
6052:
5985:, which yield an infinite family of
5685:) can be done efficiently for prime
5256:
1529:{\displaystyle ({\tfrac {4}{15}})=1}
1488:{\displaystyle ({\tfrac {2}{15}})=1}
1386:{\displaystyle ({\tfrac {a}{m}})=1,}
1245:{\displaystyle ({\tfrac {np}{p}})=0}
1003:, which is defined for all integers
10940:
10680:Lemmermeyer, pp 6–8, p. 16 ff
5539:is not congruent to 2 modulo 4, or
5385:is not congruent to 2 modulo 4 and
5114:
5107:
5012:
5005:
4914:
4673:mod the second, ..., there will be
4523:/2) minus the number in the range (
4499:
4406:The best unconditional estimate is
2187:)) if and only if (at least one of
1973:{\displaystyle {\frac {q(q-1)}{4}}}
1427:{\displaystyle ({\tfrac {a}{m}})=1}
1282:is composite and the Jacobi symbol
684:Composite modulus not a prime power
657:obey the same rules as they do mod
351:+ 1)/2 residues (including 0) and (
90:
52:; i.e., if there exists an integer
13:
11290:
11244:Stangl, Walter D. (October 1996),
10851:, pp. 49–51, (conjectured by
5950:Applications of quadratic residues
5245:and "lifted" to a solution modulo
4655:In general if a composite modulus
4569:Complexity of finding square roots
1701:
1552:Distribution of quadratic residues
1142:
1097:
510:7 ≡ 9 ≡ 49 ≡ 17
14:
14808:
11683:
11618:
11361:has been translated from Gauss's
11099:is the number of primes dividing
11060:. World Scientific. p. 250.
6022:) has been used for constructing
5488:If the complete factorization of
5241:, a solution may be found modulo
5180:are nonresidues, picking numbers
4561:congruent to 3 mod 4, the excess
3462:{\displaystyle ({\tfrac {a}{p}})}
2652:Pairs of residues and nonresidues
1656:a complex variable, and define a
1599:such that the numbers 1, 2, ...,
1569:
813:under multiplication, called the
544:is a residue modulo any power of
540:relatively prime to an odd prime
14761:
14760:
14241:
14240:
13748:
11990:
11980:
11970:
11960:
11951:
11950:
6155:; the output from this test is "
5912:The number of quadratic residues
5843:is divisible by 4 but not 8; or
4375:The least quadratic residue mod
1879:− 1 is a negative number.
518:0 ≡ 8 ≡ 16 ≡ 0
332:
11338:
11322:
11310:
11284:
11237:
11222:
11211:
11202:
11186:
11170:
11161:
11106:
11082:
11046:
11006:
10946:
10933:
10921:
10904:
10888:
10879:
10870:
10858:
10842:
10830:
10821:
10808:
10795:
10786:
10777:
10768:
10755:
7762:, 1, 4, 5, 6, 7, 9, 11, 16, 17
6086:Solovay–Strassen primality test
6009:
5969:
5862:is known. Also notice that if
5543:is congruent to 2 modulo 4 and
5485:, there may or may not be one.
5432:is congruent to 2 modulo 4 and
5325:is congruent to 2 modulo 4 and
4907:
3473:mimic a random variable like a
3414:) can be solved if and only if
2535:≡ 1, 3, 9, 19, 25, 27 (mod 28)
1928:> 3. In contrast, for prime
1871:Therefore, in this case (prime
688:The basic fact in this case is
83:
11729:analytic theory of L-functions
11707:non-abelian class field theory
11408:Gauss, Carl Friedrich (1965),
11386:Gauss, Carl Friedrich (1986),
10746:
10725:
10716:
10683:
10674:
10665:
6214:The following table (sequence
6153:generalized Riemann hypothesis
6040:Goldwasser-Micali cryptosystem
5697:, which is not known to be as
5321:then there is no solution; if
5118:
5108:
5093:
5081:
5063:
5051:
5016:
5006:
4991:
4979:
4918:
4908:
4893:
4881:
4434:Generalised Riemann hypothesis
4069:
4063:
4010:generalized Riemann hypothesis
3737:
3718:
3641:
3635:
3562:
3556:
3456:
3441:
2942:
2924:
2867:
2857:
2816:
2806:
2257:
2247:
2006:
2000:
1961:
1949:
1776:
1770:
1679:
1673:
1636:'s work (in the 1830s) on the
1634:Peter Gustav Lejeune Dirichlet
1565:
1517:
1502:
1476:
1461:
1415:
1400:
1371:
1356:
1304:
1289:
1233:
1213:
1206:under multiplication. Setting
900:
879:
848:
827:
798:, 11, 12, 13, 14 (residues in
443:. This implies the following:
394:
373:
94:
84:
1:
13977:Horner's method of evaluation
13709:History of mathematical logic
11350:
11344:Gauss, DA, arts 329–334
11056:; Diamond, Harold G. (2004).
11022:American Mathematical Society
10982:American Mathematical Society
10827:Lemmermeyer, pp 111–end
8444:Quadratic Residues (see also
6108:is composite; if they agree,
6036:quadratic residuosity problem
5695:quadratic residuosity problem
355:− 1)/2 nonresidues, by
14565:Eigenvalues and eigenvectors
14037:
13634:Primitive recursive function
11753:Transcendental number theory
11610:10.1016/0022-0000(78)90044-2
11472:Multiplicative Number Theory
11388:Disquisitiones Arithemeticae
10783:e.g. Hardy and Wright use it
10649:Law of quadratic reciprocity
7607:, 1, 2, 4, 8, 9, 13, 15, 16
6181:law of quadratic reciprocity
5954:
5772:is solvable if and only if:
5727:In general, to determine if
4942:found a similar solution if
4749:Prime or prime power modulus
4666:roots modulo the first one,
2150:Law of quadratic reciprocity
1585:law of quadratic reciprocity
958:
425:law of quadratic reciprocity
423:The first supplement to the
7:
14286:
13982:Polynomial identity testing
11976:List of recreational topics
11748:Computational number theory
11733:probabilistic number theory
11569:Lemmermeyer, Franz (2000),
11358:Disquisitiones Arithmeticae
10622:
6228:means it is not coprime to
6210:Table of quadratic residues
6177:Disquisitiones Arithmeticae
6149:Miller–Rabin primality test
6116:at least 1/2 the values of
6088:for whether a given number
5924:= 1, 2, 3 ..., looks like:
5822:for all odd prime divisors
5708:less than some given limit
4371:Least quadratic non-residue
2398:is a quadratic residue mod
2383:is a quadratic residue mod
2183:is a quadratic residue mod
2175:is a quadratic residue mod
1161: does not divide
1116: does not divide
251:(or in the list 0, 1, ...,
179:Disquisitiones Arithmeticae
10:
14813:
12698:Schröder–Bernstein theorem
12425:Monadic predicate calculus
12084:Foundations of mathematics
11533:(fifth ed.), Oxford:
11470:Davenport, Harold (2000),
11377:, the investigations into
11218:Manders & Adleman 1978
5253:or an algorithm of Gauss.
4753:First off, if the modulus
4454:showed that the number of
3685:in any interval of length
3469:for consecutive values of
3237:For example: (residues in
2195:is congruent to 1 mod 4).
2153:
645:Modulo an odd prime power
147:factoring of large numbers
14756:
14725:
14709:
14638:
14545:
14484:
14405:
14312:
14294:
14236:
14203:
14172:
14122:
14070:
14045:
13954:
13893:
13806:
13744:
13731:Philosophy of mathematics
13680:Automated theorem proving
13662:
13557:
13389:
13282:
13134:
12851:
12827:
12805:Von Neumann–Bernays–Gödel
12750:
12644:
12548:
12446:
12437:
12364:
12299:
12205:
12127:
12044:
11946:
11928:Diophantine approximation
11900:
11887:Chinese remainder theorem
11809:
11691:
11381:, and unpublished notes.
11298:. BBC Research Department
11115:, p. 113; computing
10898:(pp 511–533 of the
6406:, 10, 13, 16, 19, 22, 25
6192:continued fraction method
6128:is composite"; for prime
6120:in the range 2, 3, ...,
5718:fixed-parameter tractable
4699:Chinese remainder theorem
3756:It is easy to prove that
2625:≡ 1, 3, 4, 5, 9 (mod 11)
1753:Dirichlet showed that if
1589:Chinese remainder theorem
11772:Arithmetic combinatorics
11231:Elementary Number Theory
10763:Ireland & Rosen 1990
10659:
5999:≡ 3 (mod 4), that yield
5966:and quadratic residues.
5221:Tonelli–Shanks algorithm
4573:That is, given a number
4246:had proved in 1932 that
4135:had proved in 1918 that
2515:≡ 1, 5, 19, 23 (mod 24)
1883:For example, modulo 11,
1603:are all residues modulo
1595:> 0 there are primes
861:, and the squares are a
524:4 ≡ 12 ≡ 16.
295:{\displaystyle \rfloor }
272:{\displaystyle \lfloor }
14639:Algebraic constructions
14342:Algebraic number theory
14223:List of geometry topics
13967:Greatest common divisor
13381:Self-verifying theories
13202:Tarski's axiomatization
12153:Tarski's undefinability
12148:incompleteness theorems
11743:Geometric number theory
11699:Algebraic number theory
11535:Oxford University Press
11379:biquadratic reciprocity
11329:Bach & Shallit 1996
11317:Bach & Shallit 1996
11193:Bach & Shallit 1996
11177:Bach & Shallit 1996
11113:Bach & Shallit 1996
11089:Bach & Shallit 1996
6289:quadratic residues mod
6278:quadratic residues mod
6267:quadratic residues mod
6124:− 1 will return "
4624:) roots (i.e. zero if
2712:= 0, 1 define the sets
2524:≡ 1, 5, 7, 11 (mod 24)
1581:arithmetic progressions
1434:we do not know whether
731:is a nonresidue modulo
723:is a nonresidue modulo
620:is a nonresidue modulo
586:is a nonresidue modulo
507:5 ≡ 11 ≡ 25
129:Originally an abstract
14382:Noncommutative algebra
13839:Quadratic function (2)
13755:Mathematics portal
13366:Proof of impossibility
13014:propositional variable
12324:Propositional calculus
11862:Transcendental numbers
11776:additive number theory
11725:Analytic number theory
11456:, New York: Springer,
11246:"Counting Squares in ℤ
11229:Burton, David (2007).
11144:
11058:Analytic Number Theory
10855:, proved by Dirichlet)
10792:Gauss, DA, art 230 ff.
10654:Quadratic residue code
5816:
5655:
5641:suffices to factorize
5590:
5529:
5479:
5422:
5375:
5315:
5213:
5151:
4929:
4842:
4789:
4746:
4735:Solve x ≡ 7 (mod 15).
4722:Solve x ≡ 4 (mod 15).
4709:Solve x ≡ 6 (mod 15).
4612:, a quadratic residue
4508:be an odd prime. The
4354:
4291:
4233:
4180:
4122:
4059:
3992:
3909:
3853:
3803:
3747:
3672:
3609:
3552:
3463:
3400:
3227:
3105:
2953:
2888:
2660:, the number of pairs
2504:≡ 1, 3, 7, 9 (mod 20)
2344:
2306:
2168:are odd primes, then:
2144:
2125:
2096:
1974:
1918:
1862:
1823:
1744:
1705:
1625:
1530:
1489:
1428:
1387:
1323:
1246:
1178:
907:
855:
807:
772:
527:
514:and the even ones are
504:3 ≡ 13 ≡ 9
501:1 ≡ 15 ≡ 1
401:
296:
273:
139:acoustical engineering
105:
14619:Orthogonal complement
14392:Representation theory
14195:Differential geometry
13822:Constant function (0)
13624:Kolmogorov complexity
13577:Computably enumerable
13477:Model complete theory
13269:Principia Mathematica
12329:Propositional formula
12158:Banach–Tarski paradox
11933:Irrationality measure
11923:Diophantine equations
11766:Hodge–Arakelov theory
11587:Manders, Kenneth L.;
11145:
10956:, 27:9–16, 1987
7232:, 1, 3, 4, 9, 10, 12
6204:congruence of squares
6171:Integer factorization
6038:is the basis for the
6024:cryptographic schemes
6003:conference matrices.
5817:
5716:; however, this is a
5614:congruence of squares
5606:
5598:integer factorization
5591:
5530:
5480:
5423:
5376:
5316:
5214:
5152:
4930:
4843:
4799:using a variation of
4790:
4703:
4355:
4271:
4234:
4160:
4123:
4027:
3993:
3877:
3854:
3771:
3748:
3673:
3610:
3520:
3464:
3422: + 64
3235:
3228:
3106:
2954:
2889:
2704:be an odd prime. For
2345:
2307:
2156:quadratic reciprocity
2140:
2126:
2063:
1975:
1881:
1863:
1797:
1745:
1685:
1609:
1531:
1490:
1429:
1388:
1324:
1260:of all the integers.
1247:
1179:
908:
856:
776:
749:
739:prime power dividing
712:prime power dividing
495:
477:is an odd number and
402:
297:
274:
106:
14797:NP-complete problems
14717:Algebraic structures
14485:Algebraic structures
14470:Equivalence relation
14413:Algebraic expression
13955:Tools and algorithms
13875:Quintic function (5)
13863:Quartic function (4)
13800:polynomial functions
13572:Church–Turing thesis
13559:Computability theory
12768:continuum hypothesis
12286:Square of opposition
12144:Gödel's completeness
11892:Arithmetic functions
11758:Diophantine geometry
11573:, Berlin: Springer,
11432:Efficient Algorithms
11259:Mathematics Magazine
11119:
10970:Friedlander, John B.
5946:is given by Stangl.
5783:
5657:Determining whether
5547:
5496:
5436:
5389:
5329:
5279:
5188:
4953:
4863:
4811:
4764:
4565:is always positive.
4363:for infinitely many
4253:
4142:
4019:
3869:
3763:
3696:
3629:
3512:
3438:
3125:
3006:
2904:
2719:
2319:
2205:
1994:
1940:
1764:
1667:
1658:Dirichlet L-function
1628:Dirichlet's formulas
1499:
1458:
1397:
1353:
1286:
1210:
1021:
876:
824:
704:is a residue modulo
696:is a residue modulo
601:is a residue modulo
572:is a residue modulo
485:is a residue modulo
370:
286:
263:
255:) is symmetric (mod
120:quadratic nonresidue
63:
14646:Composition algebra
14570:Inner product space
14548:multilinear algebra
14436:Polynomial function
14377:Multilinear algebra
14362:Homological algebra
14352:Commutative algebra
13885:Septic equation (7)
13880:Sextic equation (6)
13827:Linear function (1)
13726:Mathematical object
13617:P versus NP problem
13582:Computable function
13376:Reverse mathematics
13302:Logical consequence
13179:primitive recursive
13174:elementary function
12947:Free/bound variable
12800:Tarski–Grothendieck
12319:Logical connectives
12249:Logical equivalence
12099:Logical consequence
11938:Continued fractions
11801:Arithmetic dynamics
11796:Arithmetic topology
11790:P-adic Hodge theory
11782:Arithmetic geometry
11715:Iwasawa–Tate theory
11630:"Quadratic Residue"
11014:Montgomery, Hugh L.
10774:Gauss, DA, art. 131
10752:Gauss, DA, art. 102
10743:Gauss, DA, art. 101
10731:Gauss, DA, art. 103
8453:
5990:conference matrices
4585:to tell whether an
3487:Dirichlet character
1652:be a prime number,
1577:Dirichlet's theorem
1071: divides
1001:quadratic character
469:Prime power modulus
14792:Modular arithmetic
14426:Quadratic equation
14357:Elementary algebra
14325:Algebraic geometry
14190:Algebraic geometry
13851:Cubic function (3)
13844:Quadratic equation
13524:Transfer principle
13487:Semantics of logic
13472:Categorical theory
13448:Non-standard model
12962:Logical connective
12089:Information theory
12038:Mathematical logic
11882:Modular arithmetic
11852:Irrational numbers
11786:anabelian geometry
11703:class field theory
11627:Weisstein, Eric W.
11514:A7.1: AN1, pg.249.
11167:Lemmermeyer, p. 29
11140:
10722:Gauss, DA, art 111
10713:Gauss, DA, art. 98
10701:Gauss, DA, art. 96
10689:Gauss, DA, art. 94
8443:
7213:, 41, 45, 47, 49,
6648:, 26, 31, 34, 36,
6200:number field sieve
6047:discrete logarithm
6032:oblivious transfer
6028:Rabin cryptosystem
5854:is divisible by 8.
5812:
5800:
5724:is the parameter.
5712:, this problem is
5586:
5574:
5525:
5513:
5492:is not known, and
5475:
5463:
5418:
5406:
5371:
5356:
5311:
5296:
5209:
5147:
5142:
4925:
4838:
4801:Euclid's algorithm
4785:
4436:, Ankeny obtained
4350:
4265:
4229:
4154:
4118:
3988:
3849:
3743:
3668:
3605:
3459:
3454:
3351:, 12, 13, 14, 15,
3223:
3101:
2949:
2884:
2544:≡ 1, 2, 4 (mod 7)
2365:that don't divide
2357:Thus, for numbers
2340:
2302:
2179:) if and only if (
2121:
1970:
1906:, 10 (residues in
1858:
1757:≡ 3 (mod 4), then
1740:
1526:
1515:
1485:
1474:
1424:
1413:
1383:
1369:
1319:
1302:
1242:
1231:
1174:
1169:
999:, also called the
903:
851:
555:If the modulus is
397:
292:
269:
135:modular arithmetic
101:
14787:Quadratic residue
14774:
14773:
14696:Symmetric algebra
14666:Geometric algebra
14446:Linear inequality
14397:Universal algebra
14330:Algebraic variety
14254:
14253:
14005:
14004:
13946:Quasi-homogeneous
13762:
13761:
13694:Abstract category
13497:Theories of truth
13307:Rule of inference
13297:Natural deduction
13278:
13277:
12823:
12822:
12528:Cartesian product
12433:
12432:
12339:Many-valued logic
12314:Boolean functions
12197:Russell's paradox
12172:diagonal argument
12069:First-order logic
12004:
12003:
11901:Advanced concepts
11857:Algebraic numbers
11842:Composite numbers
11544:978-0-19-853171-5
11503:, W. H. Freeman,
11493:Johnson, David S.
11489:Garey, Michael R.
11134:
10991:978-0-8218-4970-5
10671:Lemmemeyer, Ch. 1
10639:Zolotarev's lemma
10629:Euler's criterion
10620:
10619:
8441:
8440:
6188:Dixon's algorithm
6084:) correctly. The
6058:Euler's criterion
6053:Primality testing
5983:conference graphs
5799:
5693:, this forms the
5573:
5512:
5462:
5405:
5355:
5295:
5257:Composite modulus
5203:
5135:
5127:
5033:
5025:
4826:
4805:Euler's criterion
4779:
4686:... roots modulo
4581:, how hard is it
4333:
4326:
4304:
4256:
4227:
4220:
4193:
4145:
4093:
3980:
3958:
3951:
3922:
3835:
3816:
3726:
3707:
3659:
3586:
3497:and any integers
3453:
3406:≡ 1 (mod 4) then
3218:
3157:
3099:
3038:
2848:
2797:
2649:
2648:
2636:≡ 1, 11 (mod 12)
2454:≡ 1, 11 (mod 12)
2334:
2298:
2277:
2238:
2220:
2109:
2094:
2061:
2058:
2041:
1968:
1846:
1832:
1795:
1794:
1718:
1514:
1473:
1412:
1368:
1301:
1230:
1162:
1154:
1137:
1117:
1109:
1092:
1072:
1064:
1036:
1007:and positive odd
763:, 5 (residues in
669:in the product ≥
357:Euler's criterion
32:quadratic residue
14804:
14764:
14763:
14651:Exterior algebra
14320:Abstract algebra
14281:
14274:
14267:
14258:
14257:
14244:
14243:
14032:
14025:
14018:
14009:
14008:
13868:Quartic equation
13789:
13782:
13775:
13766:
13765:
13753:
13752:
13704:History of logic
13699:Category of sets
13592:Decision problem
13371:Ordinal analysis
13312:Sequent calculus
13210:Boolean algebras
13150:
13149:
13124:
13095:logical/constant
12849:
12848:
12835:
12758:Zermelo–Fraenkel
12509:Set operations:
12444:
12443:
12381:
12212:
12211:
12192:Löwenheim–Skolem
12079:Formal semantics
12031:
12024:
12017:
12008:
12007:
11994:
11984:
11974:
11973:
11964:
11963:
11954:
11953:
11847:Rational numbers
11678:
11671:
11664:
11655:
11654:
11640:
11639:
11614:
11612:
11589:Adleman, Leonard
11583:
11565:
11547:
11532:
11513:
11502:
11484:
11466:
11448:
11426:
11404:
11363:Ciceronian Latin
11345:
11342:
11336:
11326:
11320:
11314:
11308:
11307:
11305:
11303:
11297:
11288:
11282:
11281:
11256:
11241:
11235:
11234:
11226:
11220:
11215:
11209:
11206:
11200:
11190:
11184:
11174:
11168:
11165:
11159:
11149:
11147:
11146:
11141:
11139:
11135:
11127:
11110:
11104:
11086:
11080:
11079:
11054:Bateman, Paul T.
11050:
11044:
11043:
11010:
11004:
11003:
10966:
10957:
10954:J. Number Theory
10950:
10944:
10937:
10931:
10925:
10919:
10908:
10902:
10892:
10886:
10883:
10877:
10874:
10868:
10862:
10856:
10846:
10840:
10834:
10828:
10825:
10819:
10812:
10806:
10799:
10793:
10790:
10784:
10781:
10775:
10772:
10766:
10759:
10753:
10750:
10744:
10741:
10732:
10729:
10723:
10720:
10714:
10711:
10702:
10699:
10690:
10687:
10681:
10678:
10672:
10669:
8454:
8442:
8437:
8433:
8429:
8425:
8421:
8417:
8413:
8409:
8405:
8401:
8397:
8393:
8384:
8380:
8376:
8372:
8368:
8364:
8360:
8356:
8352:
8348:
8344:
8340:
8331:
8320:
8316:
8312:
8308:
8304:
8300:
8296:
8292:
8288:
8284:
8280:
8276:
8272:
8268:
8264:
8260:
8256:
8252:
8248:
8244:
8235:
8227:
8223:
8219:
8215:
8211:
8200:
8192:
8188:
8184:
8180:
8176:
8172:
8163:
8153:
8149:
8145:
8141:
8137:
8133:
8129:
8125:
8121:
8112:
8104:
8100:
8096:
8092:
8088:
8084:
8080:
8069:
8060:
8056:
8052:
8048:
8044:
8040:
8036:
8032:
8028:
8024:
8020:
8016:
8012:
8004:
8000:
7996:
7992:
7988:
7978:
7974:
7970:
7966:
7962:
7958:
7954:
7950:
7946:
7942:
7938:
7934:
7930:
7926:
7922:
7918:
7914:
7910:
7902:
7898:
7894:
7890:
7886:
7882:
7874:
7870:
7866:
7862:
7851:
7847:
7843:
7839:
7835:
7831:
7827:
7823:
7819:
7815:
7811:
7807:
7803:
7794:
7790:
7786:
7782:
7778:
7774:
7770:
7761:
7751:
7747:
7743:
7739:
7735:
7731:
7727:
7723:
7719:
7715:
7706:
7698:
7694:
7690:
7686:
7682:
7671:
7663:
7659:
7655:
7651:
7647:
7643:
7639:
7635:
7631:
7627:
7623:
7619:
7615:
7606:
7596:
7592:
7588:
7584:
7580:
7576:
7572:
7568:
7564:
7560:
7556:
7552:
7548:
7544:
7540:
7536:
7532:
7528:
7524:
7515:
7506:
7502:
7491:
7487:
7483:
7479:
7475:
7471:
7467:
7463:
7459:
7451:
7447:
7443:
7439:
7435:
7431:
7427:
7419:
7415:
7411:
7407:
7396:
7392:
7388:
7384:
7376:
7372:
7368:
7364:
7360:
7356:
7352:
7348:
7339:
7335:
7331:
7327:
7323:
7312:
7308:
7304:
7300:
7296:
7292:
7288:
7280:
7276:
7272:
7268:
7264:
7260:
7256:
7252:
7248:
7244:
7240:
7231:
7220:
7216:
7212:
7208:
7204:
7200:
7196:
7192:
7188:
7184:
7180:
7176:
7172:
7168:
7164:
7160:
7156:
7147:
7139:
7135:
7131:
7120:
7112:
7108:
7104:
7100:
7096:
7088:, 1, 3, 4, 5, 9
7087:
7076:
7072:
7068:
7064:
7060:
7056:
7052:
7048:
7044:
7040:
7032:
7028:
7024:
7020:
7016:
7012:
7003:
6999:
6995:
6991:
6980:
6971:
6967:
6963:
6959:
6955:
6951:
6947:
6943:
6939:
6935:
6926:
6915:
6911:
6907:
6903:
6899:
6895:
6891:
6887:
6883:
6879:
6875:
6871:
6867:
6863:
6859:
6855:
6846:
6842:
6838:
6834:
6830:
6826:
6822:
6814:
6810:
6799:
6795:
6791:
6787:
6783:
6779:
6775:
6771:
6767:
6763:
6759:
6750:
6746:
6742:
6733:
6723:
6719:
6715:
6711:
6707:
6703:
6699:
6695:
6691:
6682:
6674:
6670:
6666:
6655:
6651:
6647:
6643:
6639:
6635:
6631:
6627:
6619:
6615:
6611:
6607:
6603:
6599:
6595:
6591:
6587:
6583:
6574:
6564:
6560:
6556:
6552:
6548:
6544:
6540:
6536:
6532:
6528:
6524:
6520:
6516:
6507:
6498:
6487:
6478:
6474:
6470:
6466:
6462:
6453:
6442:
6438:
6434:
6430:
6426:
6422:
6418:
6414:
6405:
6401:
6392:
6381:
6377:
6373:
6369:
6365:
6361:
6357:
6353:
6349:
6345:
6336:
6332:
6328:
6324:
6320:
6316:
6312:
6308:
6259:
6258:
6255:
6245:
6227:
6219:
5933:
5903:
5876:
5849:
5838:
5821:
5819:
5818:
5813:
5805:
5801:
5792:
5771:
5756:
5744:
5684:
5674:
5640:
5630:
5595:
5593:
5592:
5587:
5579:
5575:
5572:
5568:
5556:
5534:
5532:
5531:
5526:
5518:
5514:
5505:
5484:
5482:
5481:
5476:
5468:
5464:
5461:
5457:
5445:
5427:
5425:
5424:
5419:
5411:
5407:
5398:
5380:
5378:
5377:
5372:
5361:
5357:
5354:
5350:
5338:
5320:
5318:
5317:
5312:
5301:
5297:
5288:
5274:Kronecker symbol
5218:
5216:
5215:
5210:
5208:
5204:
5196:
5156:
5154:
5153:
5148:
5146:
5145:
5136:
5133:
5128:
5125:
5121:
5105:
5104:
5100:
5075:
5074:
5070:
5034:
5031:
5026:
5023:
5019:
5003:
5002:
4998:
4934:
4932:
4931:
4926:
4921:
4905:
4904:
4900:
4847:
4845:
4844:
4839:
4831:
4827:
4819:
4797:quickly computed
4794:
4792:
4791:
4786:
4784:
4780:
4772:
4536:
4510:quadratic excess
4500:Quadratic excess
4490:
4424:
4423:
4418:for any θ>1/4
4398:
4397:
4359:
4357:
4356:
4351:
4334:
4329:
4327:
4319:
4314:
4310:
4309:
4305:
4297:
4290:
4285:
4264:
4238:
4236:
4235:
4230:
4228:
4223:
4221:
4219:
4208:
4203:
4199:
4198:
4194:
4186:
4179:
4174:
4153:
4127:
4125:
4124:
4119:
4114:
4110:
4094:
4089:
4076:
4072:
4058:
4047:
3997:
3995:
3994:
3989:
3981:
3976:
3959:
3954:
3952:
3950:
3949:
3937:
3932:
3928:
3927:
3923:
3915:
3908:
3897:
3858:
3856:
3855:
3850:
3836:
3831:
3826:
3822:
3821:
3817:
3809:
3802:
3791:
3752:
3750:
3749:
3744:
3727:
3722:
3708:
3700:
3677:
3675:
3674:
3669:
3664:
3660:
3652:
3614:
3612:
3611:
3606:
3601:
3597:
3587:
3582:
3569:
3565:
3551:
3540:
3477:. Specifically,
3468:
3466:
3465:
3460:
3455:
3446:
3395:= {2,12,13,14}.
3377:= {1,7,9,11,17},
3232:
3230:
3229:
3224:
3219:
3214:
3203:
3198:
3197:
3185:
3184:
3172:
3171:
3158:
3153:
3142:
3137:
3136:
3110:
3108:
3107:
3102:
3100:
3095:
3084:
3079:
3078:
3066:
3065:
3053:
3052:
3039:
3034:
3023:
3018:
3017:
2958:
2956:
2955:
2950:
2945:
2940:
2939:
2927:
2919:
2918:
2893:
2891:
2890:
2885:
2880:
2876:
2875:
2874:
2853:
2849:
2844:
2833:
2824:
2823:
2802:
2798:
2790:
2734:
2733:
2372:
2371:
2349:
2347:
2346:
2341:
2339:
2335:
2327:
2311:
2309:
2308:
2303:
2301:
2300:
2299:
2294:
2283:
2278:
2273:
2262:
2243:
2239:
2231:
2225:
2221:
2213:
2130:
2128:
2127:
2122:
2114:
2110:
2102:
2095:
2090:
2079:
2077:
2062:
2060:
2059:
2054:
2051:
2047:
2046:
2042:
2034:
2013:
1979:
1977:
1976:
1971:
1969:
1964:
1944:
1867:
1865:
1864:
1859:
1851:
1847:
1839:
1833:
1825:
1822:
1811:
1796:
1790:
1786:
1749:
1747:
1746:
1741:
1736:
1735:
1723:
1719:
1711:
1704:
1699:
1638:analytic formula
1611:For example, if
1547:
1543:
1539:
1535:
1533:
1532:
1527:
1516:
1507:
1494:
1492:
1491:
1486:
1475:
1466:
1453:
1443:
1433:
1431:
1430:
1425:
1414:
1405:
1392:
1390:
1389:
1384:
1370:
1361:
1348:
1338:
1328:
1326:
1325:
1320:
1303:
1294:
1281:
1273:
1251:
1249:
1248:
1243:
1232:
1226:
1218:
1200:
1190:
1183:
1181:
1180:
1175:
1173:
1172:
1163:
1160:
1155:
1152:
1138:
1135:
1118:
1115:
1110:
1107:
1093:
1090:
1073:
1070:
1065:
1062:
1041:
1037:
1029:
1013:
1006:
990:
986:
982:
978:
970:
966:
930:relatively prime
912:
910:
909:
904:
899:
891:
886:
860:
858:
857:
852:
847:
839:
834:
678:Klein four-group
636:is a nonresidue.
406:
404:
403:
398:
393:
385:
380:
301:
299:
298:
293:
278:
276:
275:
270:
250:
199:
110:
108:
107:
102:
97:
75:
74:
14812:
14811:
14807:
14806:
14805:
14803:
14802:
14801:
14777:
14776:
14775:
14770:
14752:
14721:
14705:
14686:Quotient object
14676:Polynomial ring
14634:
14595:Linear subspace
14547:
14541:
14480:
14422:Linear equation
14401:
14347:Category theory
14308:
14290:
14285:
14255:
14250:
14232:
14199:
14168:
14125:
14118:
14073:
14066:
14041:
14036:
14006:
14001:
13950:
13889:
13832:Linear equation
13802:
13793:
13763:
13758:
13747:
13740:
13685:Category theory
13675:Algebraic logic
13658:
13629:Lambda calculus
13567:Church encoding
13553:
13529:Truth predicate
13385:
13351:Complete theory
13274:
13143:
13139:
13135:
13130:
13122:
12842: and
12838:
12833:
12819:
12795:New Foundations
12763:axiom of choice
12746:
12708:Gödel numbering
12648: and
12640:
12544:
12429:
12379:
12360:
12309:Boolean algebra
12295:
12259:Equiconsistency
12224:Classical logic
12201:
12182:Halting problem
12170: and
12146: and
12134: and
12133:
12128:Theorems (
12123:
12040:
12035:
12005:
12000:
11942:
11908:Quadratic forms
11896:
11871:P-adic analysis
11827:Natural numbers
11805:
11762:Arakelov theory
11687:
11682:
11621:
11581:
11563:
11545:
11511:
11482:
11464:
11446:
11424:
11402:
11353:
11348:
11343:
11339:
11327:
11323:
11315:
11311:
11301:
11299:
11295:
11289:
11285:
11271:10.2307/2690536
11254:
11251:
11242:
11238:
11227:
11223:
11216:
11212:
11207:
11203:
11191:
11187:
11175:
11171:
11166:
11162:
11150:requires O(log
11126:
11122:
11120:
11117:
11116:
11111:
11107:
11087:
11083:
11068:
11051:
11047:
11032:
11024:. p. 176.
11011:
11007:
10992:
10984:. p. 156.
10978:Opera De Cribro
10974:Iwaniec, Henryk
10967:
10960:
10951:
10947:
10938:
10934:
10926:
10922:
10909:
10905:
10893:
10889:
10884:
10880:
10875:
10871:
10863:
10859:
10847:
10843:
10835:
10831:
10826:
10822:
10813:
10809:
10800:
10796:
10791:
10787:
10782:
10778:
10773:
10769:
10760:
10756:
10751:
10747:
10742:
10735:
10730:
10726:
10721:
10717:
10712:
10705:
10700:
10693:
10688:
10684:
10679:
10675:
10670:
10666:
10662:
10625:
8435:
8431:
8427:
8423:
8419:
8415:
8411:
8407:
8403:
8399:
8395:
8391:
8382:
8378:
8374:
8370:
8366:
8362:
8358:
8354:
8350:
8346:
8342:
8338:
8329:
8318:
8314:
8310:
8306:
8302:
8298:
8294:
8290:
8286:
8282:
8278:
8274:
8270:
8266:
8262:
8258:
8254:
8250:
8246:
8242:
8233:
8225:
8221:
8217:
8213:
8209:
8198:
8190:
8186:
8182:
8178:
8174:
8170:
8161:
8151:
8147:
8143:
8139:
8135:
8131:
8127:
8123:
8119:
8110:
8102:
8098:
8094:
8090:
8086:
8082:
8078:
8067:
8058:
8054:
8050:
8046:
8042:
8038:
8034:
8030:
8026:
8022:
8018:
8014:
8010:
8002:
7998:
7994:
7990:
7986:
7976:
7972:
7968:
7964:
7960:
7956:
7952:
7948:
7944:
7940:
7936:
7932:
7928:
7924:
7920:
7916:
7912:
7908:
7900:
7896:
7892:
7888:
7884:
7880:
7872:
7868:
7864:
7860:
7849:
7845:
7841:
7837:
7833:
7829:
7825:
7821:
7817:
7813:
7809:
7805:
7801:
7792:
7788:
7784:
7780:
7776:
7772:
7768:
7759:
7749:
7745:
7741:
7737:
7733:
7729:
7725:
7721:
7717:
7713:
7704:
7696:
7692:
7688:
7684:
7680:
7669:
7661:
7657:
7653:
7649:
7645:
7641:
7637:
7633:
7629:
7625:
7621:
7617:
7613:
7604:
7594:
7590:
7586:
7582:
7578:
7574:
7570:
7566:
7562:
7558:
7554:
7550:
7546:
7542:
7538:
7534:
7530:
7526:
7522:
7513:
7504:
7500:
7489:
7485:
7481:
7477:
7473:
7469:
7465:
7461:
7457:
7449:
7445:
7441:
7437:
7433:
7429:
7425:
7417:
7413:
7409:
7405:
7394:
7390:
7386:
7382:
7374:
7370:
7366:
7362:
7358:
7354:
7350:
7346:
7337:
7333:
7329:
7325:
7321:
7310:
7306:
7302:
7298:
7294:
7290:
7286:
7278:
7274:
7270:
7266:
7262:
7258:
7254:
7250:
7246:
7242:
7238:
7229:
7218:
7214:
7210:
7206:
7202:
7198:
7194:
7190:
7186:
7182:
7178:
7174:
7170:
7166:
7162:
7158:
7154:
7145:
7137:
7133:
7129:
7118:
7110:
7106:
7102:
7098:
7094:
7085:
7074:
7070:
7066:
7062:
7058:
7054:
7050:
7046:
7042:
7038:
7030:
7026:
7022:
7018:
7014:
7013:, 1, 4, 9, 11,
7010:
7001:
6997:
6993:
6989:
6978:
6969:
6965:
6961:
6957:
6953:
6949:
6945:
6941:
6937:
6933:
6924:
6913:
6909:
6905:
6901:
6897:
6893:
6889:
6885:
6881:
6877:
6873:
6869:
6865:
6861:
6857:
6853:
6844:
6840:
6836:
6832:
6828:
6824:
6820:
6812:
6808:
6797:
6793:
6789:
6785:
6781:
6777:
6773:
6769:
6765:
6761:
6757:
6748:
6744:
6740:
6731:
6721:
6717:
6713:
6709:
6705:
6701:
6697:
6693:
6689:
6680:
6672:
6668:
6664:
6653:
6649:
6645:
6641:
6637:
6633:
6629:
6625:
6617:
6613:
6609:
6605:
6601:
6597:
6593:
6589:
6585:
6581:
6572:
6562:
6558:
6554:
6550:
6546:
6542:
6538:
6534:
6530:
6526:
6522:
6518:
6514:
6505:
6496:
6485:
6476:
6472:
6468:
6464:
6460:
6451:
6440:
6436:
6432:
6428:
6424:
6420:
6416:
6412:
6403:
6399:
6390:
6379:
6375:
6371:
6367:
6363:
6359:
6355:
6351:
6347:
6343:
6334:
6330:
6326:
6322:
6318:
6314:
6310:
6306:
6247:
6237:
6225:
6215:
6212:
6196:quadratic sieve
6175:In § VI of the
6173:
6055:
6012:
5972:
5964:primitive roots
5960:Sound diffusers
5957:
5952:
5929:
5914:
5878:
5863:
5844:
5833:
5790:
5786:
5784:
5781:
5780:
5778:Legendre symbol
5758:
5746:
5739:
5720:problem, where
5676:
5666:
5632:
5617:
5564:
5560:
5554:
5550:
5548:
5545:
5544:
5503:
5499:
5497:
5494:
5493:
5453:
5449:
5443:
5439:
5437:
5434:
5433:
5396:
5392:
5390:
5387:
5386:
5346:
5342:
5336:
5332:
5330:
5327:
5326:
5286:
5282:
5280:
5277:
5276:
5261:If the modulus
5259:
5226:If the modulus
5195:
5191:
5189:
5186:
5185:
5141:
5140:
5132:
5124:
5122:
5106:
5096:
5080:
5076:
5066:
5050:
5046:
5039:
5038:
5030:
5022:
5020:
5004:
4994:
4978:
4974:
4963:
4962:
4954:
4951:
4950:
4906:
4896:
4880:
4876:
4864:
4861:
4860:
4818:
4814:
4812:
4809:
4808:
4771:
4767:
4765:
4762:
4761:
4759:Legendre symbol
4751:
4685:
4679:
4672:
4665:
4571:
4532:
4502:
4486:
4477:for odd primes
4421:
4419:
4393:
4391:
4373:
4328:
4318:
4296:
4292:
4286:
4275:
4270:
4266:
4260:
4254:
4251:
4250:
4222:
4212:
4207:
4185:
4181:
4175:
4164:
4159:
4155:
4149:
4143:
4140:
4139:
4088:
4087:
4083:
4048:
4031:
4026:
4022:
4020:
4017:
4016:
3975:
3953:
3945:
3941:
3936:
3914:
3910:
3898:
3881:
3876:
3872:
3870:
3867:
3866:
3830:
3808:
3804:
3792:
3775:
3770:
3766:
3764:
3761:
3760:
3721:
3699:
3697:
3694:
3693:
3651:
3647:
3630:
3627:
3626:
3581:
3580:
3576:
3541:
3524:
3519:
3515:
3513:
3510:
3509:
3444:
3439:
3436:
3435:
3432:
3394:
3385:
3376:
3367:
3313:
3304:
3295:
3286:
3204:
3202:
3193:
3189:
3180:
3176:
3167:
3163:
3143:
3141:
3132:
3128:
3126:
3123:
3122:
3085:
3083:
3074:
3070:
3061:
3057:
3048:
3044:
3024:
3022:
3013:
3009:
3007:
3004:
3003:
2990:
2983:
2976:
2969:
2941:
2932:
2928:
2923:
2911:
2907:
2905:
2902:
2901:
2870:
2866:
2834:
2832:
2828:
2819:
2815:
2789:
2785:
2742:
2738:
2726:
2722:
2720:
2717:
2716:
2656:Modulo a prime
2654:
2564:≡ 1, 3 (mod 8)
2555:≡ 1, 7 (mod 8)
2495:≡ 1, 4 (mod 5)
2443:≡ 1, 3 (mod 8)
2434:≡ 1, 7 (mod 8)
2402:if and only if
2387:if and only if
2361:and odd primes
2352:Legendre symbol
2326:
2322:
2320:
2317:
2316:
2284:
2282:
2263:
2261:
2260:
2256:
2230:
2226:
2212:
2208:
2206:
2203:
2202:
2158:
2152:
2101:
2097:
2080:
2078:
2067:
2053:
2033:
2029:
2022:
2018:
2017:
2012:
1995:
1992:
1991:
1945:
1943:
1941:
1938:
1937:
1838:
1834:
1824:
1812:
1801:
1785:
1765:
1762:
1761:
1728:
1724:
1710:
1706:
1700:
1689:
1668:
1665:
1664:
1646:quadratic forms
1630:
1554:
1545:
1541:
1537:
1505:
1500:
1497:
1496:
1464:
1459:
1456:
1455:
1454:. For example:
1445:
1435:
1403:
1398:
1395:
1394:
1359:
1354:
1351:
1350:
1340:
1330:
1292:
1287:
1284:
1283:
1279:
1271:
1219:
1216:
1211:
1208:
1207:
1204:complex numbers
1198:
1188:
1168:
1167:
1159:
1153: and
1151:
1134:
1132:
1123:
1122:
1114:
1108: and
1106:
1089:
1087:
1078:
1077:
1069:
1061:
1059:
1046:
1045:
1028:
1024:
1022:
1019:
1018:
1011:
1004:
997:Legendre symbol
988:
984:
980:
976:
968:
964:
961:
944:if and only if
932:to the modulus
895:
887:
882:
877:
874:
873:
843:
835:
830:
825:
822:
821:
686:
489:if and only if
471:
389:
381:
376:
371:
368:
367:
335:
287:
284:
283:
264:
261:
260:
245:
194:
155:
82:
70:
66:
64:
61:
60:
17:
12:
11:
5:
14810:
14800:
14799:
14794:
14789:
14772:
14771:
14769:
14768:
14757:
14754:
14753:
14751:
14750:
14745:
14740:
14738:Linear algebra
14735:
14729:
14727:
14723:
14722:
14720:
14719:
14713:
14711:
14707:
14706:
14704:
14703:
14701:Tensor algebra
14698:
14693:
14690:Quotient group
14683:
14673:
14663:
14653:
14648:
14642:
14640:
14636:
14635:
14633:
14632:
14627:
14622:
14612:
14609:Euclidean norm
14602:
14592:
14582:
14577:
14567:
14562:
14557:
14551:
14549:
14543:
14542:
14540:
14539:
14529:
14519:
14509:
14499:
14488:
14486:
14482:
14481:
14479:
14478:
14473:
14463:
14460:Multiplication
14449:
14439:
14429:
14415:
14409:
14407:
14406:Basic concepts
14403:
14402:
14400:
14399:
14394:
14389:
14384:
14379:
14374:
14372:Linear algebra
14369:
14364:
14359:
14354:
14349:
14344:
14339:
14338:
14337:
14332:
14322:
14316:
14314:
14310:
14309:
14307:
14306:
14301:
14295:
14292:
14291:
14284:
14283:
14276:
14269:
14261:
14252:
14251:
14249:
14248:
14237:
14234:
14233:
14231:
14230:
14225:
14220:
14219:
14218:
14207:
14205:
14201:
14200:
14198:
14197:
14192:
14187:
14182:
14176:
14174:
14170:
14169:
14167:
14166:
14161:
14156:
14151:
14146:
14141:
14136:
14130:
14128:
14124:Non-Euclidean
14120:
14119:
14117:
14116:
14114:Solid geometry
14111:
14110:
14109:
14104:
14097:Plane geometry
14094:
14089:
14084:
14078:
14076:
14068:
14067:
14065:
14064:
14059:
14058:
14057:
14046:
14043:
14042:
14035:
14034:
14027:
14020:
14012:
14003:
14002:
14000:
13999:
13994:
13989:
13984:
13979:
13974:
13969:
13964:
13958:
13956:
13952:
13951:
13949:
13948:
13943:
13938:
13933:
13928:
13923:
13918:
13913:
13908:
13903:
13897:
13895:
13891:
13890:
13888:
13887:
13882:
13877:
13872:
13871:
13870:
13860:
13859:
13858:
13856:Cubic equation
13848:
13847:
13846:
13836:
13835:
13834:
13824:
13819:
13813:
13811:
13804:
13803:
13792:
13791:
13784:
13777:
13769:
13760:
13759:
13745:
13742:
13741:
13739:
13738:
13733:
13728:
13723:
13718:
13717:
13716:
13706:
13701:
13696:
13687:
13682:
13677:
13672:
13670:Abstract logic
13666:
13664:
13660:
13659:
13657:
13656:
13651:
13649:Turing machine
13646:
13641:
13636:
13631:
13626:
13621:
13620:
13619:
13614:
13609:
13604:
13599:
13589:
13587:Computable set
13584:
13579:
13574:
13569:
13563:
13561:
13555:
13554:
13552:
13551:
13546:
13541:
13536:
13531:
13526:
13521:
13516:
13515:
13514:
13509:
13504:
13494:
13489:
13484:
13482:Satisfiability
13479:
13474:
13469:
13468:
13467:
13457:
13456:
13455:
13445:
13444:
13443:
13438:
13433:
13428:
13423:
13413:
13412:
13411:
13406:
13399:Interpretation
13395:
13393:
13387:
13386:
13384:
13383:
13378:
13373:
13368:
13363:
13353:
13348:
13347:
13346:
13345:
13344:
13334:
13329:
13319:
13314:
13309:
13304:
13299:
13294:
13288:
13286:
13280:
13279:
13276:
13275:
13273:
13272:
13264:
13263:
13262:
13261:
13256:
13255:
13254:
13249:
13244:
13224:
13223:
13222:
13220:minimal axioms
13217:
13206:
13205:
13204:
13193:
13192:
13191:
13186:
13181:
13176:
13171:
13166:
13153:
13151:
13132:
13131:
13129:
13128:
13127:
13126:
13114:
13109:
13108:
13107:
13102:
13097:
13092:
13082:
13077:
13072:
13067:
13066:
13065:
13060:
13050:
13049:
13048:
13043:
13038:
13033:
13023:
13018:
13017:
13016:
13011:
13006:
12996:
12995:
12994:
12989:
12984:
12979:
12974:
12969:
12959:
12954:
12949:
12944:
12943:
12942:
12937:
12932:
12927:
12917:
12912:
12910:Formation rule
12907:
12902:
12901:
12900:
12895:
12885:
12884:
12883:
12873:
12868:
12863:
12858:
12852:
12846:
12829:Formal systems
12825:
12824:
12821:
12820:
12818:
12817:
12812:
12807:
12802:
12797:
12792:
12787:
12782:
12777:
12772:
12771:
12770:
12765:
12754:
12752:
12748:
12747:
12745:
12744:
12743:
12742:
12732:
12727:
12726:
12725:
12718:Large cardinal
12715:
12710:
12705:
12700:
12695:
12681:
12680:
12679:
12674:
12669:
12654:
12652:
12642:
12641:
12639:
12638:
12637:
12636:
12631:
12626:
12616:
12611:
12606:
12601:
12596:
12591:
12586:
12581:
12576:
12571:
12566:
12561:
12555:
12553:
12546:
12545:
12543:
12542:
12541:
12540:
12535:
12530:
12525:
12520:
12515:
12507:
12506:
12505:
12500:
12490:
12485:
12483:Extensionality
12480:
12478:Ordinal number
12475:
12465:
12460:
12459:
12458:
12447:
12441:
12435:
12434:
12431:
12430:
12428:
12427:
12422:
12417:
12412:
12407:
12402:
12397:
12396:
12395:
12385:
12384:
12383:
12370:
12368:
12362:
12361:
12359:
12358:
12357:
12356:
12351:
12346:
12336:
12331:
12326:
12321:
12316:
12311:
12305:
12303:
12297:
12296:
12294:
12293:
12288:
12283:
12278:
12273:
12268:
12263:
12262:
12261:
12251:
12246:
12241:
12236:
12231:
12226:
12220:
12218:
12209:
12203:
12202:
12200:
12199:
12194:
12189:
12184:
12179:
12174:
12162:Cantor's
12160:
12155:
12150:
12140:
12138:
12125:
12124:
12122:
12121:
12116:
12111:
12106:
12101:
12096:
12091:
12086:
12081:
12076:
12071:
12066:
12061:
12060:
12059:
12048:
12046:
12042:
12041:
12034:
12033:
12026:
12019:
12011:
12002:
12001:
11999:
11998:
11988:
11978:
11968:
11966:List of topics
11958:
11947:
11944:
11943:
11941:
11940:
11935:
11930:
11925:
11920:
11915:
11910:
11904:
11902:
11898:
11897:
11895:
11894:
11889:
11884:
11879:
11874:
11867:P-adic numbers
11864:
11859:
11854:
11849:
11844:
11839:
11834:
11829:
11824:
11819:
11813:
11811:
11807:
11806:
11804:
11803:
11798:
11793:
11779:
11769:
11755:
11750:
11745:
11740:
11722:
11711:Iwasawa theory
11695:
11693:
11689:
11688:
11681:
11680:
11673:
11666:
11658:
11652:
11651:
11641:
11620:
11619:External links
11617:
11616:
11615:
11603:(2): 168–184,
11584:
11579:
11566:
11561:
11548:
11543:
11515:
11509:
11485:
11480:
11467:
11462:
11449:
11444:
11427:
11422:
11405:
11400:
11352:
11349:
11347:
11346:
11337:
11321:
11309:
11283:
11265:(4): 285–289,
11247:
11236:
11221:
11210:
11201:
11185:
11169:
11160:
11138:
11133:
11130:
11125:
11105:
11095:) steps where
11081:
11066:
11045:
11030:
11005:
10990:
10958:
10945:
10941:external links
10932:
10928:Davenport 2000
10920:
10903:
10887:
10878:
10869:
10865:Davenport 2000
10857:
10849:Davenport 2000
10841:
10837:Davenport 2000
10829:
10820:
10807:
10794:
10785:
10776:
10767:
10754:
10745:
10733:
10724:
10715:
10703:
10691:
10682:
10673:
10663:
10661:
10658:
10657:
10656:
10651:
10646:
10641:
10636:
10631:
10624:
10621:
10618:
10617:
10614:
10611:
10608:
10605:
10602:
10599:
10596:
10593:
10590:
10587:
10584:
10581:
10578:
10575:
10572:
10569:
10566:
10563:
10560:
10557:
10554:
10551:
10548:
10545:
10542:
10538:
10537:
10534:
10531:
10528:
10525:
10522:
10519:
10516:
10513:
10510:
10507:
10504:
10501:
10498:
10495:
10492:
10489:
10486:
10483:
10480:
10477:
10474:
10471:
10468:
10465:
10462:
10458:
10457:
10454:
10451:
10448:
10445:
10442:
10439:
10436:
10433:
10430:
10427:
10424:
10421:
10418:
10415:
10412:
10409:
10406:
10403:
10400:
10397:
10394:
10391:
10388:
10385:
10382:
10378:
10377:
10374:
10371:
10368:
10365:
10362:
10359:
10356:
10353:
10350:
10347:
10344:
10341:
10338:
10335:
10332:
10329:
10326:
10323:
10320:
10317:
10314:
10311:
10308:
10305:
10302:
10298:
10297:
10294:
10291:
10288:
10285:
10282:
10279:
10276:
10273:
10270:
10267:
10264:
10261:
10258:
10255:
10252:
10249:
10246:
10243:
10240:
10237:
10234:
10231:
10228:
10225:
10222:
10218:
10217:
10214:
10211:
10208:
10205:
10202:
10199:
10196:
10193:
10190:
10187:
10184:
10181:
10178:
10175:
10172:
10169:
10166:
10163:
10160:
10157:
10154:
10151:
10148:
10145:
10142:
10138:
10137:
10134:
10131:
10128:
10125:
10122:
10119:
10116:
10113:
10110:
10107:
10104:
10101:
10098:
10095:
10092:
10089:
10086:
10083:
10080:
10077:
10074:
10071:
10068:
10065:
10062:
10058:
10057:
10054:
10051:
10048:
10045:
10042:
10039:
10036:
10033:
10030:
10027:
10024:
10021:
10018:
10015:
10012:
10009:
10006:
10003:
10000:
9997:
9994:
9991:
9988:
9985:
9982:
9978:
9977:
9974:
9971:
9968:
9965:
9962:
9959:
9956:
9953:
9950:
9947:
9944:
9941:
9938:
9935:
9932:
9929:
9926:
9923:
9920:
9917:
9914:
9911:
9908:
9905:
9902:
9898:
9897:
9894:
9891:
9888:
9885:
9882:
9879:
9876:
9873:
9870:
9867:
9864:
9861:
9858:
9855:
9852:
9849:
9846:
9843:
9840:
9837:
9834:
9831:
9828:
9825:
9822:
9818:
9817:
9814:
9811:
9808:
9805:
9802:
9799:
9796:
9793:
9790:
9787:
9784:
9781:
9778:
9775:
9772:
9769:
9766:
9763:
9760:
9757:
9754:
9751:
9748:
9745:
9742:
9738:
9737:
9734:
9731:
9728:
9725:
9722:
9719:
9716:
9713:
9710:
9707:
9704:
9701:
9698:
9695:
9692:
9689:
9686:
9683:
9680:
9677:
9674:
9671:
9668:
9665:
9662:
9658:
9657:
9654:
9651:
9648:
9645:
9642:
9639:
9636:
9633:
9630:
9627:
9624:
9621:
9618:
9615:
9612:
9609:
9606:
9603:
9600:
9597:
9594:
9591:
9588:
9585:
9582:
9578:
9577:
9574:
9571:
9568:
9565:
9562:
9559:
9556:
9553:
9550:
9547:
9544:
9541:
9538:
9535:
9532:
9529:
9526:
9523:
9520:
9517:
9514:
9511:
9508:
9505:
9502:
9498:
9497:
9494:
9491:
9488:
9485:
9482:
9479:
9476:
9473:
9470:
9467:
9464:
9461:
9458:
9455:
9452:
9449:
9446:
9443:
9440:
9437:
9434:
9431:
9428:
9425:
9422:
9418:
9417:
9414:
9411:
9408:
9405:
9402:
9399:
9396:
9393:
9390:
9387:
9384:
9381:
9378:
9375:
9372:
9369:
9366:
9363:
9360:
9357:
9354:
9351:
9348:
9345:
9342:
9338:
9337:
9334:
9331:
9328:
9325:
9322:
9319:
9316:
9313:
9310:
9307:
9304:
9301:
9298:
9295:
9292:
9289:
9286:
9283:
9280:
9277:
9274:
9271:
9268:
9265:
9262:
9258:
9257:
9254:
9251:
9248:
9245:
9242:
9239:
9236:
9233:
9230:
9227:
9224:
9221:
9218:
9215:
9212:
9209:
9206:
9203:
9200:
9197:
9194:
9191:
9188:
9185:
9182:
9178:
9177:
9174:
9171:
9168:
9165:
9162:
9159:
9156:
9153:
9150:
9147:
9144:
9141:
9138:
9135:
9132:
9129:
9126:
9123:
9120:
9117:
9114:
9111:
9108:
9105:
9102:
9098:
9097:
9094:
9091:
9088:
9085:
9082:
9079:
9076:
9073:
9070:
9067:
9064:
9061:
9058:
9055:
9052:
9049:
9046:
9043:
9040:
9037:
9034:
9031:
9028:
9025:
9022:
9018:
9017:
9014:
9011:
9008:
9005:
9002:
8999:
8996:
8993:
8990:
8987:
8984:
8981:
8978:
8975:
8972:
8969:
8966:
8963:
8960:
8957:
8954:
8951:
8948:
8945:
8942:
8938:
8937:
8934:
8931:
8928:
8925:
8922:
8919:
8916:
8913:
8910:
8907:
8904:
8901:
8898:
8895:
8892:
8889:
8886:
8883:
8880:
8877:
8874:
8871:
8868:
8865:
8862:
8858:
8857:
8854:
8851:
8848:
8845:
8842:
8839:
8836:
8833:
8830:
8827:
8824:
8821:
8818:
8815:
8812:
8809:
8806:
8803:
8800:
8797:
8794:
8791:
8788:
8785:
8782:
8778:
8777:
8774:
8771:
8768:
8765:
8762:
8759:
8756:
8753:
8750:
8747:
8744:
8741:
8738:
8735:
8732:
8729:
8726:
8723:
8720:
8717:
8714:
8711:
8708:
8705:
8702:
8698:
8697:
8694:
8691:
8688:
8685:
8682:
8679:
8676:
8673:
8670:
8667:
8664:
8661:
8658:
8655:
8652:
8649:
8646:
8643:
8640:
8637:
8634:
8631:
8628:
8625:
8622:
8618:
8617:
8614:
8611:
8608:
8605:
8602:
8599:
8596:
8593:
8590:
8587:
8584:
8581:
8578:
8575:
8572:
8569:
8566:
8563:
8560:
8557:
8554:
8551:
8548:
8545:
8542:
8536:
8535:
8532:
8529:
8526:
8523:
8520:
8517:
8514:
8511:
8508:
8505:
8502:
8499:
8496:
8493:
8490:
8487:
8484:
8481:
8478:
8475:
8472:
8469:
8466:
8463:
8460:
8439:
8438:
8389:
8386:
8336:
8333:
8327:
8323:
8322:
8240:
8237:
8231:
8228:
8207:
8203:
8202:
8196:
8193:
8168:
8165:
8159:
8155:
8154:
8117:
8114:
8108:
8105:
8076:
8072:
8071:
8065:
8062:
8053:, 27, 29, 31,
8008:
8005:
7984:
7980:
7979:
7906:
7903:
7878:
7875:
7858:
7854:
7853:
7799:
7796:
7766:
7763:
7757:
7753:
7752:
7711:
7708:
7702:
7699:
7678:
7674:
7673:
7667:
7664:
7611:
7608:
7602:
7598:
7597:
7520:
7517:
7511:
7508:
7498:
7494:
7493:
7455:
7452:
7423:
7420:
7403:
7399:
7398:
7393:, 17, 25, 33,
7380:
7377:
7365:, 16, 22, 25,
7344:
7341:
7319:
7315:
7314:
7309:, 37, 43, 46,
7284:
7281:
7236:
7233:
7227:
7223:
7222:
7152:
7149:
7143:
7140:
7127:
7123:
7122:
7116:
7113:
7092:
7089:
7083:
7079:
7078:
7036:
7033:
7008:
7005:
6987:
6983:
6982:
6976:
6973:
6960:, 19, 21, 25,
6931:
6928:
6922:
6918:
6917:
6908:, 45, 49, 51,
6851:
6848:
6818:
6815:
6806:
6802:
6801:
6755:
6752:
6738:
6735:
6729:
6725:
6724:
6687:
6684:
6678:
6675:
6662:
6658:
6657:
6623:
6620:
6579:
6576:
6570:
6566:
6565:
6512:
6509:
6503:
6500:
6494:
6490:
6489:
6483:
6480:
6458:
6455:
6449:
6445:
6444:
6431:, 17, 25, 29,
6410:
6407:
6397:
6394:
6388:
6384:
6383:
6341:
6338:
6304:
6301:
6298:
6294:
6293:
6287:
6282:
6276:
6271:
6265:
6211:
6208:
6172:
6169:
6142:probable prime
6096:and computes (
6054:
6051:
6011:
6008:
5971:
5968:
5956:
5953:
5951:
5948:
5940:
5939:
5913:
5910:
5908:is a square).
5856:
5855:
5831:
5811:
5808:
5804:
5798:
5795:
5789:
5585:
5582:
5578:
5571:
5567:
5563:
5559:
5553:
5524:
5521:
5517:
5511:
5508:
5502:
5474:
5471:
5467:
5460:
5456:
5452:
5448:
5442:
5417:
5414:
5410:
5404:
5401:
5395:
5370:
5367:
5364:
5360:
5353:
5349:
5345:
5341:
5335:
5310:
5307:
5304:
5300:
5294:
5291:
5285:
5258:
5255:
5251:Hensel's lemma
5207:
5202:
5199:
5194:
5168:(in 1891) and
5158:
5157:
5144:
5139:
5131:
5126: if
5123:
5120:
5117:
5113:
5110:
5103:
5099:
5095:
5092:
5089:
5086:
5083:
5079:
5073:
5069:
5065:
5062:
5059:
5056:
5053:
5049:
5044:
5041:
5040:
5037:
5029:
5024: if
5021:
5018:
5015:
5011:
5008:
5001:
4997:
4993:
4990:
4987:
4984:
4981:
4977:
4972:
4969:
4968:
4966:
4961:
4958:
4936:
4935:
4924:
4920:
4917:
4913:
4910:
4903:
4899:
4895:
4892:
4889:
4886:
4883:
4879:
4874:
4871:
4868:
4837:
4834:
4830:
4825:
4822:
4817:
4783:
4778:
4775:
4770:
4750:
4747:
4745:
4744:
4743:
4742:
4739:
4732:
4731:
4730:
4729:
4726:
4719:
4718:
4717:
4716:
4713:
4697:is called the
4683:
4677:
4670:
4663:
4606:
4605:
4602:
4577:and a modulus
4570:
4567:
4501:
4498:
4497:
4496:
4427:character sums
4372:
4369:
4361:
4360:
4349:
4346:
4343:
4340:
4337:
4332:
4325:
4322:
4317:
4313:
4308:
4303:
4300:
4295:
4289:
4284:
4281:
4278:
4274:
4269:
4263:
4259:
4240:
4239:
4226:
4218:
4215:
4211:
4206:
4202:
4197:
4192:
4189:
4184:
4178:
4173:
4170:
4167:
4163:
4158:
4152:
4148:
4129:
4128:
4117:
4113:
4109:
4106:
4103:
4100:
4097:
4092:
4086:
4082:
4079:
4075:
4071:
4068:
4065:
4062:
4057:
4054:
4051:
4046:
4043:
4040:
4037:
4034:
4030:
4025:
3999:
3998:
3987:
3984:
3979:
3974:
3971:
3968:
3965:
3962:
3957:
3948:
3944:
3940:
3935:
3931:
3926:
3921:
3918:
3913:
3907:
3904:
3901:
3896:
3893:
3890:
3887:
3884:
3880:
3875:
3860:
3859:
3848:
3845:
3842:
3839:
3834:
3829:
3825:
3820:
3815:
3812:
3807:
3801:
3798:
3795:
3790:
3787:
3784:
3781:
3778:
3774:
3769:
3754:
3753:
3742:
3739:
3736:
3733:
3730:
3725:
3720:
3717:
3714:
3711:
3706:
3703:
3679:
3678:
3667:
3663:
3658:
3655:
3650:
3646:
3643:
3640:
3637:
3634:
3620:big O notation
3616:
3615:
3604:
3600:
3596:
3593:
3590:
3585:
3579:
3575:
3572:
3568:
3564:
3561:
3558:
3555:
3550:
3547:
3544:
3539:
3536:
3533:
3530:
3527:
3523:
3518:
3458:
3452:
3449:
3443:
3434:The values of
3431:
3428:
3399:
3398:
3397:
3396:
3392:
3387:
3386:= {3,8,10,15},
3383:
3378:
3374:
3369:
3365:
3318:
3317:
3316:
3315:
3314:= {5,6,10,11}.
3311:
3306:
3305:= {3,7,12,14},
3302:
3297:
3293:
3288:
3284:
3267:, 10, 11, 12,
3234:
3233:
3222:
3217:
3213:
3210:
3207:
3201:
3196:
3192:
3188:
3183:
3179:
3175:
3170:
3166:
3161:
3156:
3152:
3149:
3146:
3140:
3135:
3131:
3112:
3111:
3098:
3094:
3091:
3088:
3082:
3077:
3073:
3069:
3064:
3060:
3056:
3051:
3047:
3042:
3037:
3033:
3030:
3027:
3021:
3016:
3012:
2993:
2992:
2988:
2985:
2981:
2978:
2974:
2971:
2967:
2960:
2959:
2948:
2944:
2938:
2935:
2931:
2926:
2922:
2917:
2914:
2910:
2895:
2894:
2883:
2879:
2873:
2869:
2865:
2862:
2859:
2856:
2852:
2847:
2843:
2840:
2837:
2831:
2827:
2822:
2818:
2814:
2811:
2808:
2805:
2801:
2796:
2793:
2788:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2741:
2737:
2732:
2729:
2725:
2653:
2650:
2647:
2646:
2640:
2637:
2631:
2627:
2626:
2620:
2617:
2611:
2607:
2606:
2600:
2597:
2591:
2587:
2586:
2580:
2577:
2570:
2566:
2565:
2559:
2556:
2550:
2546:
2545:
2539:
2536:
2530:
2526:
2525:
2519:
2516:
2510:
2506:
2505:
2499:
2496:
2490:
2486:
2485:
2479:
2476:
2469:
2465:
2464:
2458:
2455:
2449:
2445:
2444:
2438:
2435:
2429:
2425:
2424:
2418:
2415:
2408:
2404:
2403:
2393:
2388:
2378:
2338:
2333:
2330:
2325:
2313:
2312:
2297:
2293:
2290:
2287:
2281:
2276:
2272:
2269:
2266:
2259:
2255:
2252:
2249:
2246:
2242:
2237:
2234:
2229:
2224:
2219:
2216:
2211:
2154:Main article:
2151:
2148:
2138:− 1)/2.
2132:
2131:
2120:
2117:
2113:
2108:
2105:
2100:
2093:
2089:
2086:
2083:
2076:
2073:
2070:
2066:
2057:
2050:
2045:
2040:
2037:
2032:
2028:
2025:
2021:
2016:
2011:
2008:
2005:
2002:
1999:
1967:
1963:
1960:
1957:
1954:
1951:
1948:
1917:
1916:
1912:
1911:
1869:
1868:
1857:
1854:
1850:
1845:
1842:
1837:
1831:
1828:
1821:
1818:
1815:
1810:
1807:
1804:
1800:
1793:
1789:
1784:
1781:
1778:
1775:
1772:
1769:
1751:
1750:
1739:
1734:
1731:
1727:
1722:
1717:
1714:
1709:
1703:
1698:
1695:
1692:
1688:
1684:
1681:
1678:
1675:
1672:
1629:
1626:
1553:
1550:
1525:
1522:
1519:
1513:
1510:
1504:
1484:
1481:
1478:
1472:
1469:
1463:
1423:
1420:
1417:
1411:
1408:
1402:
1382:
1379:
1376:
1373:
1367:
1364:
1358:
1318:
1315:
1312:
1309:
1306:
1300:
1297:
1291:
1241:
1238:
1235:
1229:
1225:
1222:
1215:
1185:
1184:
1171:
1166:
1158:
1150:
1147:
1144:
1141:
1136: if
1133:
1131:
1128:
1125:
1124:
1121:
1113:
1105:
1102:
1099:
1096:
1091: if
1088:
1086:
1083:
1080:
1079:
1076:
1068:
1063: if
1060:
1058:
1052:
1051:
1049:
1044:
1040:
1035:
1032:
1027:
993:
992:
960:
957:
948:is coprime to
940:is coprime to
917:for composite
902:
898:
894:
890:
885:
881:
850:
846:
842:
838:
833:
829:
815:group of units
745:
744:
717:
685:
682:
640:
639:
638:
637:
618:
599:
584:
526:
525:
522:
519:
512:
511:
508:
505:
502:
470:
467:
396:
392:
388:
384:
379:
375:
340:Modulo an odd
334:
331:
310:cannot exceed
291:
268:
154:
151:
112:
111:
100:
96:
93:
89:
86:
81:
78:
73:
69:
46:perfect square
15:
9:
6:
4:
3:
2:
14809:
14798:
14795:
14793:
14790:
14788:
14785:
14784:
14782:
14767:
14759:
14758:
14755:
14749:
14746:
14744:
14741:
14739:
14736:
14734:
14731:
14730:
14728:
14724:
14718:
14715:
14714:
14712:
14708:
14702:
14699:
14697:
14694:
14691:
14687:
14684:
14681:
14677:
14674:
14671:
14667:
14664:
14661:
14657:
14654:
14652:
14649:
14647:
14644:
14643:
14641:
14637:
14631:
14628:
14626:
14623:
14620:
14616:
14615:Orthogonality
14613:
14610:
14606:
14603:
14600:
14596:
14593:
14590:
14586:
14583:
14581:
14580:Hilbert space
14578:
14575:
14571:
14568:
14566:
14563:
14561:
14558:
14556:
14553:
14552:
14550:
14544:
14537:
14533:
14530:
14527:
14523:
14520:
14517:
14513:
14510:
14507:
14503:
14500:
14497:
14493:
14490:
14489:
14487:
14483:
14477:
14474:
14471:
14467:
14464:
14461:
14457:
14453:
14450:
14447:
14443:
14440:
14437:
14433:
14430:
14427:
14423:
14419:
14416:
14414:
14411:
14410:
14408:
14404:
14398:
14395:
14393:
14390:
14388:
14385:
14383:
14380:
14378:
14375:
14373:
14370:
14368:
14365:
14363:
14360:
14358:
14355:
14353:
14350:
14348:
14345:
14343:
14340:
14336:
14333:
14331:
14328:
14327:
14326:
14323:
14321:
14318:
14317:
14315:
14311:
14305:
14302:
14300:
14297:
14296:
14293:
14289:
14282:
14277:
14275:
14270:
14268:
14263:
14262:
14259:
14247:
14239:
14238:
14235:
14229:
14226:
14224:
14221:
14217:
14214:
14213:
14212:
14209:
14208:
14206:
14202:
14196:
14193:
14191:
14188:
14186:
14183:
14181:
14178:
14177:
14175:
14171:
14165:
14162:
14160:
14157:
14155:
14152:
14150:
14147:
14145:
14142:
14140:
14137:
14135:
14132:
14131:
14129:
14127:
14121:
14115:
14112:
14108:
14105:
14103:
14100:
14099:
14098:
14095:
14093:
14090:
14088:
14085:
14083:
14082:Combinatorial
14080:
14079:
14077:
14075:
14069:
14063:
14060:
14056:
14053:
14052:
14051:
14048:
14047:
14044:
14040:
14033:
14028:
14026:
14021:
14019:
14014:
14013:
14010:
13998:
13997:Gröbner basis
13995:
13993:
13990:
13988:
13985:
13983:
13980:
13978:
13975:
13973:
13970:
13968:
13965:
13963:
13962:Factorization
13960:
13959:
13957:
13953:
13947:
13944:
13942:
13939:
13937:
13934:
13932:
13929:
13927:
13924:
13922:
13919:
13917:
13914:
13912:
13909:
13907:
13904:
13902:
13899:
13898:
13896:
13894:By properties
13892:
13886:
13883:
13881:
13878:
13876:
13873:
13869:
13866:
13865:
13864:
13861:
13857:
13854:
13853:
13852:
13849:
13845:
13842:
13841:
13840:
13837:
13833:
13830:
13829:
13828:
13825:
13823:
13820:
13818:
13815:
13814:
13812:
13810:
13805:
13801:
13797:
13790:
13785:
13783:
13778:
13776:
13771:
13770:
13767:
13757:
13756:
13751:
13743:
13737:
13734:
13732:
13729:
13727:
13724:
13722:
13719:
13715:
13712:
13711:
13710:
13707:
13705:
13702:
13700:
13697:
13695:
13691:
13688:
13686:
13683:
13681:
13678:
13676:
13673:
13671:
13668:
13667:
13665:
13661:
13655:
13652:
13650:
13647:
13645:
13644:Recursive set
13642:
13640:
13637:
13635:
13632:
13630:
13627:
13625:
13622:
13618:
13615:
13613:
13610:
13608:
13605:
13603:
13600:
13598:
13595:
13594:
13593:
13590:
13588:
13585:
13583:
13580:
13578:
13575:
13573:
13570:
13568:
13565:
13564:
13562:
13560:
13556:
13550:
13547:
13545:
13542:
13540:
13537:
13535:
13532:
13530:
13527:
13525:
13522:
13520:
13517:
13513:
13510:
13508:
13505:
13503:
13500:
13499:
13498:
13495:
13493:
13490:
13488:
13485:
13483:
13480:
13478:
13475:
13473:
13470:
13466:
13463:
13462:
13461:
13458:
13454:
13453:of arithmetic
13451:
13450:
13449:
13446:
13442:
13439:
13437:
13434:
13432:
13429:
13427:
13424:
13422:
13419:
13418:
13417:
13414:
13410:
13407:
13405:
13402:
13401:
13400:
13397:
13396:
13394:
13392:
13388:
13382:
13379:
13377:
13374:
13372:
13369:
13367:
13364:
13361:
13360:from ZFC
13357:
13354:
13352:
13349:
13343:
13340:
13339:
13338:
13335:
13333:
13330:
13328:
13325:
13324:
13323:
13320:
13318:
13315:
13313:
13310:
13308:
13305:
13303:
13300:
13298:
13295:
13293:
13290:
13289:
13287:
13285:
13281:
13271:
13270:
13266:
13265:
13260:
13259:non-Euclidean
13257:
13253:
13250:
13248:
13245:
13243:
13242:
13238:
13237:
13235:
13232:
13231:
13229:
13225:
13221:
13218:
13216:
13213:
13212:
13211:
13207:
13203:
13200:
13199:
13198:
13194:
13190:
13187:
13185:
13182:
13180:
13177:
13175:
13172:
13170:
13167:
13165:
13162:
13161:
13159:
13155:
13154:
13152:
13147:
13141:
13136:Example
13133:
13125:
13120:
13119:
13118:
13115:
13113:
13110:
13106:
13103:
13101:
13098:
13096:
13093:
13091:
13088:
13087:
13086:
13083:
13081:
13078:
13076:
13073:
13071:
13068:
13064:
13061:
13059:
13056:
13055:
13054:
13051:
13047:
13044:
13042:
13039:
13037:
13034:
13032:
13029:
13028:
13027:
13024:
13022:
13019:
13015:
13012:
13010:
13007:
13005:
13002:
13001:
13000:
12997:
12993:
12990:
12988:
12985:
12983:
12980:
12978:
12975:
12973:
12970:
12968:
12965:
12964:
12963:
12960:
12958:
12955:
12953:
12950:
12948:
12945:
12941:
12938:
12936:
12933:
12931:
12928:
12926:
12923:
12922:
12921:
12918:
12916:
12913:
12911:
12908:
12906:
12903:
12899:
12896:
12894:
12893:by definition
12891:
12890:
12889:
12886:
12882:
12879:
12878:
12877:
12874:
12872:
12869:
12867:
12864:
12862:
12859:
12857:
12854:
12853:
12850:
12847:
12845:
12841:
12836:
12830:
12826:
12816:
12813:
12811:
12808:
12806:
12803:
12801:
12798:
12796:
12793:
12791:
12788:
12786:
12783:
12781:
12780:Kripke–Platek
12778:
12776:
12773:
12769:
12766:
12764:
12761:
12760:
12759:
12756:
12755:
12753:
12749:
12741:
12738:
12737:
12736:
12733:
12731:
12728:
12724:
12721:
12720:
12719:
12716:
12714:
12711:
12709:
12706:
12704:
12701:
12699:
12696:
12693:
12689:
12685:
12682:
12678:
12675:
12673:
12670:
12668:
12665:
12664:
12663:
12659:
12656:
12655:
12653:
12651:
12647:
12643:
12635:
12632:
12630:
12627:
12625:
12624:constructible
12622:
12621:
12620:
12617:
12615:
12612:
12610:
12607:
12605:
12602:
12600:
12597:
12595:
12592:
12590:
12587:
12585:
12582:
12580:
12577:
12575:
12572:
12570:
12567:
12565:
12562:
12560:
12557:
12556:
12554:
12552:
12547:
12539:
12536:
12534:
12531:
12529:
12526:
12524:
12521:
12519:
12516:
12514:
12511:
12510:
12508:
12504:
12501:
12499:
12496:
12495:
12494:
12491:
12489:
12486:
12484:
12481:
12479:
12476:
12474:
12470:
12466:
12464:
12461:
12457:
12454:
12453:
12452:
12449:
12448:
12445:
12442:
12440:
12436:
12426:
12423:
12421:
12418:
12416:
12413:
12411:
12408:
12406:
12403:
12401:
12398:
12394:
12391:
12390:
12389:
12386:
12382:
12377:
12376:
12375:
12372:
12371:
12369:
12367:
12363:
12355:
12352:
12350:
12347:
12345:
12342:
12341:
12340:
12337:
12335:
12332:
12330:
12327:
12325:
12322:
12320:
12317:
12315:
12312:
12310:
12307:
12306:
12304:
12302:
12301:Propositional
12298:
12292:
12289:
12287:
12284:
12282:
12279:
12277:
12274:
12272:
12269:
12267:
12264:
12260:
12257:
12256:
12255:
12252:
12250:
12247:
12245:
12242:
12240:
12237:
12235:
12232:
12230:
12229:Logical truth
12227:
12225:
12222:
12221:
12219:
12217:
12213:
12210:
12208:
12204:
12198:
12195:
12193:
12190:
12188:
12185:
12183:
12180:
12178:
12175:
12173:
12169:
12165:
12161:
12159:
12156:
12154:
12151:
12149:
12145:
12142:
12141:
12139:
12137:
12131:
12126:
12120:
12117:
12115:
12112:
12110:
12107:
12105:
12102:
12100:
12097:
12095:
12092:
12090:
12087:
12085:
12082:
12080:
12077:
12075:
12072:
12070:
12067:
12065:
12062:
12058:
12055:
12054:
12053:
12050:
12049:
12047:
12043:
12039:
12032:
12027:
12025:
12020:
12018:
12013:
12012:
12009:
11997:
11993:
11989:
11987:
11983:
11979:
11977:
11969:
11967:
11959:
11957:
11949:
11948:
11945:
11939:
11936:
11934:
11931:
11929:
11926:
11924:
11921:
11919:
11916:
11914:
11913:Modular forms
11911:
11909:
11906:
11905:
11903:
11899:
11893:
11890:
11888:
11885:
11883:
11880:
11878:
11875:
11872:
11868:
11865:
11863:
11860:
11858:
11855:
11853:
11850:
11848:
11845:
11843:
11840:
11838:
11837:Prime numbers
11835:
11833:
11830:
11828:
11825:
11823:
11820:
11818:
11815:
11814:
11812:
11808:
11802:
11799:
11797:
11794:
11791:
11787:
11783:
11780:
11777:
11773:
11770:
11767:
11763:
11759:
11756:
11754:
11751:
11749:
11746:
11744:
11741:
11738:
11734:
11730:
11726:
11723:
11720:
11719:Kummer theory
11716:
11712:
11708:
11704:
11700:
11697:
11696:
11694:
11690:
11686:
11685:Number theory
11679:
11674:
11672:
11667:
11665:
11660:
11659:
11656:
11649:
11645:
11642:
11637:
11636:
11631:
11628:
11623:
11622:
11611:
11606:
11602:
11598:
11594:
11590:
11585:
11582:
11580:3-540-66957-4
11576:
11572:
11567:
11564:
11562:0-387-97329-X
11558:
11554:
11549:
11546:
11540:
11536:
11531:
11530:
11524:
11523:Wright, E. M.
11520:
11516:
11512:
11510:0-7167-1045-5
11506:
11501:
11500:
11494:
11490:
11486:
11483:
11481:0-387-95097-4
11477:
11473:
11468:
11465:
11463:0-387-94777-9
11459:
11455:
11450:
11447:
11445:0-262-02405-5
11441:
11437:
11436:The MIT Press
11433:
11428:
11425:
11423:0-8284-0191-8
11419:
11415:
11411:
11406:
11403:
11401:0-387-96254-9
11397:
11393:
11389:
11384:
11383:
11382:
11380:
11376:
11372:
11368:
11364:
11360:
11359:
11341:
11334:
11330:
11325:
11319:, p. 113
11318:
11313:
11294:
11287:
11280:
11276:
11272:
11268:
11264:
11260:
11253:
11250:
11240:
11232:
11225:
11219:
11214:
11205:
11198:
11194:
11189:
11182:
11178:
11173:
11164:
11157:
11153:
11136:
11131:
11128:
11123:
11114:
11109:
11102:
11098:
11094:
11090:
11085:
11077:
11073:
11069:
11067:981-256-080-7
11063:
11059:
11055:
11049:
11041:
11037:
11033:
11031:0-8218-0737-4
11027:
11023:
11019:
11015:
11009:
11001:
10997:
10993:
10987:
10983:
10979:
10975:
10971:
10965:
10963:
10955:
10949:
10942:
10936:
10929:
10924:
10917:
10913:
10907:
10901:
10897:
10891:
10882:
10873:
10866:
10861:
10854:
10850:
10845:
10838:
10833:
10824:
10817:
10811:
10804:
10798:
10789:
10780:
10771:
10764:
10758:
10749:
10740:
10738:
10728:
10719:
10710:
10708:
10698:
10696:
10686:
10677:
10668:
10664:
10655:
10652:
10650:
10647:
10645:
10644:Character sum
10642:
10640:
10637:
10635:
10634:Gauss's lemma
10632:
10630:
10627:
10626:
10615:
10612:
10609:
10606:
10603:
10600:
10597:
10594:
10591:
10588:
10585:
10582:
10579:
10576:
10573:
10570:
10567:
10564:
10561:
10558:
10555:
10552:
10549:
10546:
10543:
10540:
10539:
10535:
10532:
10529:
10526:
10523:
10520:
10517:
10514:
10511:
10508:
10505:
10502:
10499:
10496:
10493:
10490:
10487:
10484:
10481:
10478:
10475:
10472:
10469:
10466:
10463:
10460:
10459:
10455:
10452:
10449:
10446:
10443:
10440:
10437:
10434:
10431:
10428:
10425:
10422:
10419:
10416:
10413:
10410:
10407:
10404:
10401:
10398:
10395:
10392:
10389:
10386:
10383:
10380:
10379:
10375:
10372:
10369:
10366:
10363:
10360:
10357:
10354:
10351:
10348:
10345:
10342:
10339:
10336:
10333:
10330:
10327:
10324:
10321:
10318:
10315:
10312:
10309:
10306:
10303:
10300:
10299:
10295:
10292:
10289:
10286:
10283:
10280:
10277:
10274:
10271:
10268:
10265:
10262:
10259:
10256:
10253:
10250:
10247:
10244:
10241:
10238:
10235:
10232:
10229:
10226:
10223:
10220:
10219:
10215:
10212:
10209:
10206:
10203:
10200:
10197:
10194:
10191:
10188:
10185:
10182:
10179:
10176:
10173:
10170:
10167:
10164:
10161:
10158:
10155:
10152:
10149:
10146:
10143:
10140:
10139:
10135:
10132:
10129:
10126:
10123:
10120:
10117:
10114:
10111:
10108:
10105:
10102:
10099:
10096:
10093:
10090:
10087:
10084:
10081:
10078:
10075:
10072:
10069:
10066:
10063:
10060:
10059:
10055:
10052:
10049:
10046:
10043:
10040:
10037:
10034:
10031:
10028:
10025:
10022:
10019:
10016:
10013:
10010:
10007:
10004:
10001:
9998:
9995:
9992:
9989:
9986:
9983:
9980:
9979:
9975:
9972:
9969:
9966:
9963:
9960:
9957:
9954:
9951:
9948:
9945:
9942:
9939:
9936:
9933:
9930:
9927:
9924:
9921:
9918:
9915:
9912:
9909:
9906:
9903:
9900:
9899:
9895:
9892:
9889:
9886:
9883:
9880:
9877:
9874:
9871:
9868:
9865:
9862:
9859:
9856:
9853:
9850:
9847:
9844:
9841:
9838:
9835:
9832:
9829:
9826:
9823:
9820:
9819:
9815:
9812:
9809:
9806:
9803:
9800:
9797:
9794:
9791:
9788:
9785:
9782:
9779:
9776:
9773:
9770:
9767:
9764:
9761:
9758:
9755:
9752:
9749:
9746:
9743:
9740:
9739:
9735:
9732:
9729:
9726:
9723:
9720:
9717:
9714:
9711:
9708:
9705:
9702:
9699:
9696:
9693:
9690:
9687:
9684:
9681:
9678:
9675:
9672:
9669:
9666:
9663:
9660:
9659:
9655:
9652:
9649:
9646:
9643:
9640:
9637:
9634:
9631:
9628:
9625:
9622:
9619:
9616:
9613:
9610:
9607:
9604:
9601:
9598:
9595:
9592:
9589:
9586:
9583:
9580:
9579:
9575:
9572:
9569:
9566:
9563:
9560:
9557:
9554:
9551:
9548:
9545:
9542:
9539:
9536:
9533:
9530:
9527:
9524:
9521:
9518:
9515:
9512:
9509:
9506:
9503:
9500:
9499:
9495:
9492:
9489:
9486:
9483:
9480:
9477:
9474:
9471:
9468:
9465:
9462:
9459:
9456:
9453:
9450:
9447:
9444:
9441:
9438:
9435:
9432:
9429:
9426:
9423:
9420:
9419:
9415:
9412:
9409:
9406:
9403:
9400:
9397:
9394:
9391:
9388:
9385:
9382:
9379:
9376:
9373:
9370:
9367:
9364:
9361:
9358:
9355:
9352:
9349:
9346:
9343:
9340:
9339:
9335:
9332:
9329:
9326:
9323:
9320:
9317:
9314:
9311:
9308:
9305:
9302:
9299:
9296:
9293:
9290:
9287:
9284:
9281:
9278:
9275:
9272:
9269:
9266:
9263:
9260:
9259:
9255:
9252:
9249:
9246:
9243:
9240:
9237:
9234:
9231:
9228:
9225:
9222:
9219:
9216:
9213:
9210:
9207:
9204:
9201:
9198:
9195:
9192:
9189:
9186:
9183:
9180:
9179:
9175:
9172:
9169:
9166:
9163:
9160:
9157:
9154:
9151:
9148:
9145:
9142:
9139:
9136:
9133:
9130:
9127:
9124:
9121:
9118:
9115:
9112:
9109:
9106:
9103:
9100:
9099:
9095:
9092:
9089:
9086:
9083:
9080:
9077:
9074:
9071:
9068:
9065:
9062:
9059:
9056:
9053:
9050:
9047:
9044:
9041:
9038:
9035:
9032:
9029:
9026:
9023:
9020:
9019:
9015:
9012:
9009:
9006:
9003:
9000:
8997:
8994:
8991:
8988:
8985:
8982:
8979:
8976:
8973:
8970:
8967:
8964:
8961:
8958:
8955:
8952:
8949:
8946:
8943:
8940:
8939:
8935:
8932:
8929:
8926:
8923:
8920:
8917:
8914:
8911:
8908:
8905:
8902:
8899:
8896:
8893:
8890:
8887:
8884:
8881:
8878:
8875:
8872:
8869:
8866:
8863:
8860:
8859:
8855:
8852:
8849:
8846:
8843:
8840:
8837:
8834:
8831:
8828:
8825:
8822:
8819:
8816:
8813:
8810:
8807:
8804:
8801:
8798:
8795:
8792:
8789:
8786:
8783:
8780:
8779:
8775:
8772:
8769:
8766:
8763:
8760:
8757:
8754:
8751:
8748:
8745:
8742:
8739:
8736:
8733:
8730:
8727:
8724:
8721:
8718:
8715:
8712:
8709:
8706:
8703:
8700:
8699:
8695:
8692:
8689:
8686:
8683:
8680:
8677:
8674:
8671:
8668:
8665:
8662:
8659:
8656:
8653:
8650:
8647:
8644:
8641:
8638:
8635:
8632:
8629:
8626:
8623:
8620:
8619:
8615:
8612:
8609:
8606:
8603:
8600:
8597:
8594:
8591:
8588:
8585:
8582:
8579:
8576:
8573:
8570:
8567:
8564:
8561:
8558:
8555:
8552:
8549:
8546:
8543:
8541:
8538:
8537:
8533:
8530:
8527:
8524:
8521:
8518:
8515:
8512:
8509:
8506:
8503:
8500:
8497:
8494:
8491:
8488:
8485:
8482:
8479:
8476:
8473:
8470:
8467:
8464:
8461:
8459:
8456:
8455:
8451:
8447:
8390:
8387:
8337:
8334:
8328:
8325:
8324:
8241:
8238:
8232:
8229:
8208:
8205:
8204:
8197:
8194:
8169:
8166:
8160:
8157:
8156:
8118:
8115:
8109:
8106:
8077:
8074:
8073:
8066:
8063:
8009:
8006:
7985:
7982:
7981:
7907:
7904:
7879:
7876:
7859:
7856:
7855:
7852:, 55, 58, 64
7800:
7797:
7767:
7764:
7758:
7755:
7754:
7712:
7709:
7703:
7700:
7679:
7676:
7675:
7668:
7665:
7612:
7609:
7603:
7600:
7599:
7521:
7518:
7512:
7509:
7499:
7496:
7495:
7492:, 56, 61, 64
7456:
7453:
7424:
7421:
7404:
7401:
7400:
7397:, 41, 49, 57
7381:
7378:
7345:
7342:
7320:
7317:
7316:
7285:
7282:
7237:
7234:
7228:
7225:
7224:
7153:
7150:
7144:
7141:
7128:
7125:
7124:
7117:
7114:
7093:
7090:
7084:
7081:
7080:
7037:
7034:
7009:
7006:
6988:
6985:
6984:
6977:
6974:
6948:, 9, 13, 15,
6932:
6929:
6923:
6920:
6919:
6852:
6849:
6819:
6816:
6807:
6804:
6803:
6756:
6753:
6739:
6736:
6730:
6727:
6726:
6688:
6685:
6679:
6676:
6663:
6660:
6659:
6624:
6621:
6580:
6577:
6571:
6568:
6567:
6513:
6510:
6504:
6501:
6495:
6492:
6491:
6484:
6481:
6459:
6456:
6450:
6447:
6446:
6411:
6408:
6398:
6395:
6389:
6386:
6385:
6342:
6339:
6305:
6302:
6299:
6296:
6295:
6292:
6288:
6286:
6283:
6281:
6277:
6275:
6272:
6270:
6266:
6264:
6261:
6260:
6257:
6254:
6250:
6244:
6240:
6235:
6231:
6223:
6218:
6207:
6205:
6201:
6197:
6193:
6189:
6184:
6182:
6178:
6168:
6166:
6162:
6158:
6154:
6150:
6145:
6143:
6139:
6135:
6131:
6127:
6123:
6119:
6115:
6111:
6107:
6103:
6099:
6095:
6091:
6087:
6083:
6079:
6075:
6072:is prime. If
6071:
6067:
6063:
6059:
6050:
6048:
6043:
6041:
6037:
6033:
6029:
6025:
6021:
6017:
6007:
6004:
6002:
6001:antisymmetric
5998:
5993:
5991:
5988:
5984:
5980:
5976:
5967:
5965:
5961:
5947:
5945:
5937:
5932:
5927:
5926:
5925:
5923:
5919:
5909:
5907:
5901:
5897:
5893:
5889:
5885:
5881:
5875:
5871:
5867:
5861:
5853:
5847:
5842:
5836:
5832:
5829:
5825:
5809:
5806:
5802:
5796:
5793:
5787:
5779:
5775:
5774:
5773:
5769:
5765:
5761:
5754:
5750:
5742:
5736:
5734:
5730:
5725:
5723:
5719:
5715:
5711:
5707:
5702:
5700:
5696:
5692:
5688:
5683:
5679:
5673:
5669:
5664:
5660:
5654:
5652:
5648:
5644:
5639:
5635:
5628:
5624:
5620:
5615:
5611:
5605:
5603:
5599:
5583:
5580:
5576:
5569:
5565:
5561:
5557:
5551:
5542:
5538:
5522:
5519:
5515:
5509:
5506:
5500:
5491:
5486:
5472:
5469:
5465:
5458:
5454:
5450:
5446:
5440:
5431:
5415:
5412:
5408:
5402:
5399:
5393:
5384:
5368:
5365:
5362:
5358:
5351:
5347:
5343:
5339:
5333:
5324:
5308:
5305:
5302:
5298:
5292:
5289:
5283:
5275:
5271:
5266:
5264:
5254:
5252:
5248:
5244:
5240:
5236:
5233:
5229:
5224:
5222:
5205:
5200:
5197:
5192:
5183:
5179:
5175:
5171:
5167:
5163:
5137:
5129:
5115:
5111:
5101:
5097:
5090:
5087:
5084:
5077:
5071:
5067:
5060:
5057:
5054:
5047:
5042:
5035:
5027:
5013:
5009:
4999:
4995:
4988:
4985:
4982:
4975:
4970:
4964:
4959:
4956:
4949:
4948:
4947:
4946:≡ 5 (mod 8):
4945:
4941:
4922:
4915:
4911:
4901:
4897:
4890:
4887:
4884:
4877:
4872:
4869:
4866:
4859:
4858:
4857:
4855:
4852:≡ 3 (mod 4),
4851:
4835:
4832:
4828:
4823:
4820:
4815:
4806:
4802:
4798:
4781:
4776:
4773:
4768:
4760:
4757:is prime the
4756:
4740:
4737:
4736:
4734:
4733:
4727:
4724:
4723:
4721:
4720:
4714:
4711:
4710:
4708:
4707:
4706:
4705:For example:
4702:
4700:
4696:
4691:
4689:
4682:
4676:
4669:
4662:
4658:
4653:
4651:
4647:
4643:
4640:), or two if
4639:
4635:
4631:
4627:
4623:
4619:
4615:
4611:
4603:
4600:
4596:
4592:
4588:
4584:
4583:
4582:
4580:
4576:
4566:
4564:
4560:
4556:
4552:
4548:
4544:
4540:
4535:
4530:
4526:
4522:
4518:
4514:
4511:
4507:
4494:
4489:
4484:
4483:
4482:
4480:
4476:
4471:
4469:
4465:
4461:
4457:
4453:
4449:
4447:
4443:
4439:
4435:
4432:Assuming the
4430:
4428:
4417:
4413:
4409:
4404:
4402:
4396:
4388:
4386:
4382:
4378:
4368:
4366:
4347:
4344:
4341:
4338:
4335:
4330:
4323:
4320:
4315:
4311:
4306:
4301:
4298:
4293:
4287:
4282:
4279:
4276:
4272:
4267:
4261:
4249:
4248:
4247:
4245:
4224:
4216:
4213:
4209:
4204:
4200:
4195:
4190:
4187:
4182:
4176:
4171:
4168:
4165:
4161:
4156:
4150:
4138:
4137:
4136:
4134:
4115:
4111:
4107:
4104:
4101:
4098:
4095:
4090:
4084:
4080:
4077:
4073:
4066:
4060:
4055:
4052:
4049:
4044:
4041:
4038:
4035:
4032:
4028:
4023:
4015:
4014:
4013:
4012:is true then
4011:
4007:
4003:
3985:
3982:
3977:
3972:
3969:
3966:
3963:
3960:
3955:
3946:
3942:
3938:
3933:
3929:
3924:
3919:
3916:
3911:
3905:
3902:
3899:
3894:
3891:
3888:
3885:
3882:
3878:
3873:
3865:
3864:
3863:
3846:
3843:
3840:
3837:
3832:
3827:
3823:
3818:
3813:
3810:
3805:
3799:
3796:
3793:
3788:
3785:
3782:
3779:
3776:
3772:
3767:
3759:
3758:
3757:
3740:
3734:
3731:
3728:
3723:
3715:
3712:
3709:
3704:
3701:
3692:
3691:
3690:
3688:
3684:
3665:
3661:
3656:
3653:
3648:
3644:
3638:
3632:
3625:
3624:
3623:
3621:
3602:
3598:
3594:
3591:
3588:
3583:
3577:
3573:
3570:
3566:
3559:
3553:
3548:
3545:
3542:
3537:
3534:
3531:
3528:
3525:
3521:
3516:
3508:
3507:
3506:
3504:
3500:
3496:
3492:
3488:
3484:
3480:
3476:
3472:
3450:
3447:
3427:
3425:
3421:
3418: =
3417:
3413:
3409:
3405:
3391:
3388:
3382:
3379:
3373:
3370:
3368:= {4,5,6,16},
3364:
3361:
3360:
3358:
3354:
3350:
3346:
3342:
3338:
3334:
3330:
3326:
3323:
3322:
3321:
3310:
3307:
3301:
3298:
3296:= {2,4,9,13},
3292:
3289:
3283:
3280:
3279:
3278:
3274:
3270:
3266:
3262:
3258:
3254:
3250:
3247:
3246:
3245:
3242:
3240:
3220:
3215:
3211:
3208:
3205:
3199:
3194:
3190:
3186:
3181:
3177:
3173:
3168:
3164:
3159:
3154:
3150:
3147:
3144:
3138:
3133:
3129:
3121:
3120:
3119:
3117:
3096:
3092:
3089:
3086:
3080:
3075:
3071:
3067:
3062:
3058:
3054:
3049:
3045:
3040:
3035:
3031:
3028:
3025:
3019:
3014:
3010:
3002:
3001:
3000:
2998:
2986:
2979:
2972:
2965:
2964:
2963:
2946:
2936:
2933:
2929:
2920:
2915:
2912:
2908:
2900:
2899:
2898:
2881:
2877:
2871:
2863:
2860:
2854:
2850:
2845:
2841:
2838:
2835:
2829:
2825:
2820:
2812:
2809:
2803:
2799:
2794:
2791:
2786:
2782:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2746:
2743:
2739:
2735:
2730:
2727:
2723:
2715:
2714:
2713:
2711:
2707:
2703:
2699:
2695:
2691:
2687:
2683:
2679:
2675:
2671:
2667:
2663:
2659:
2644:
2641:
2638:
2635:
2632:
2629:
2628:
2624:
2621:
2618:
2615:
2612:
2609:
2608:
2604:
2601:
2598:
2595:
2592:
2589:
2588:
2584:
2581:
2578:
2575:
2572:(every prime
2571:
2568:
2567:
2563:
2560:
2557:
2554:
2551:
2548:
2547:
2543:
2540:
2537:
2534:
2531:
2528:
2527:
2523:
2520:
2517:
2514:
2511:
2508:
2507:
2503:
2500:
2497:
2494:
2491:
2488:
2487:
2483:
2480:
2477:
2474:
2471:(every prime
2470:
2467:
2466:
2462:
2459:
2456:
2453:
2450:
2447:
2446:
2442:
2439:
2436:
2433:
2430:
2427:
2426:
2422:
2419:
2416:
2413:
2410:(every prime
2409:
2406:
2405:
2401:
2397:
2394:
2392:
2389:
2386:
2382:
2379:
2377:
2374:
2373:
2370:
2368:
2364:
2360:
2355:
2353:
2336:
2331:
2328:
2323:
2295:
2291:
2288:
2285:
2279:
2274:
2270:
2267:
2264:
2253:
2250:
2244:
2240:
2235:
2232:
2227:
2222:
2217:
2214:
2209:
2201:
2200:
2199:
2196:
2194:
2190:
2186:
2182:
2178:
2174:
2169:
2167:
2163:
2157:
2147:
2143:
2139:
2137:
2118:
2115:
2111:
2106:
2103:
2098:
2091:
2087:
2084:
2081:
2074:
2071:
2068:
2064:
2055:
2048:
2043:
2038:
2035:
2030:
2026:
2023:
2019:
2014:
2009:
2003:
1997:
1990:
1989:
1988:
1987:≡ 3 (mod 4),
1986:
1981:
1965:
1958:
1955:
1952:
1946:
1935:
1931:
1927:
1923:
1914:
1913:
1909:
1905:
1901:
1897:
1893:
1889:
1886:
1885:
1884:
1880:
1878:
1874:
1855:
1852:
1848:
1843:
1840:
1835:
1829:
1826:
1819:
1816:
1813:
1808:
1805:
1802:
1798:
1791:
1787:
1782:
1779:
1773:
1767:
1760:
1759:
1758:
1756:
1737:
1732:
1729:
1725:
1720:
1715:
1712:
1707:
1696:
1693:
1690:
1686:
1682:
1676:
1670:
1663:
1662:
1661:
1659:
1655:
1651:
1647:
1643:
1639:
1635:
1624:
1622:
1618:
1614:
1608:
1606:
1602:
1598:
1594:
1590:
1586:
1582:
1579:on primes in
1578:
1573:
1571:
1567:
1563:
1559:
1549:
1523:
1520:
1511:
1508:
1482:
1479:
1470:
1467:
1452:
1448:
1442:
1438:
1421:
1418:
1409:
1406:
1380:
1377:
1374:
1365:
1362:
1347:
1343:
1337:
1333:
1316:
1313:
1310:
1307:
1298:
1295:
1277:
1276:Jacobi symbol
1268:
1266:
1261:
1259:
1255:
1239:
1236:
1227:
1223:
1220:
1205:
1201:
1194:
1164:
1156:
1148:
1145:
1139:
1129:
1126:
1119:
1111:
1103:
1100:
1094:
1084:
1081:
1074:
1066:
1056:
1047:
1042:
1038:
1033:
1030:
1025:
1017:
1016:
1015:
1010:
1009:prime numbers
1002:
998:
975:for example,
974:
973:
972:
956:
953:
951:
947:
943:
939:
935:
931:
927:
922:
920:
916:
915:zero divisors
892:
888:
870:
868:
864:
840:
836:
820:
816:
812:
806:
803:
801:
797:
793:
789:
785:
781:
775:
771:
768:
766:
762:
758:
754:
748:
742:
738:
734:
730:
726:
722:
718:
715:
711:
707:
703:
699:
695:
691:
690:
689:
681:
679:
674:
672:
668:
664:
660:
656:
652:
648:
643:
635:
631:
627:
623:
619:
616:
612:
608:
604:
600:
597:
593:
589:
585:
583:
579:
575:
571:
570:
569:
566:
562:
561:
560:
558:
553:
551:
547:
543:
539:
534:
532:
523:
520:
517:
516:
515:
509:
506:
503:
500:
499:
498:
494:
493:≡ 1 (mod 8).
492:
488:
484:
480:
476:
466:
464:
460:
455:
453:
449:
444:
442:
438:
434:
430:
426:
421:
418:
415:
412:
410:
386:
382:
366:
362:
358:
354:
350:
346:
343:
338:
333:Prime modulus
330:
327:
325:
321:
317:
313:
309:
305:
281:
258:
254:
248:
243:
239:
235:
231:
227:
223:
219:
215:
211:
207:
203:
197:
192:
188:
183:
181:
180:
175:
171:
167:
163:
159:
150:
148:
144:
140:
136:
132:
127:
125:
121:
118:is called a
117:
98:
91:
87:
79:
76:
71:
67:
59:
58:
57:
55:
51:
47:
43:
39:
36:
33:
29:
26:
22:
21:number theory
14743:Order theory
14733:Field theory
14599:Affine space
14532:Vector space
14387:Order theory
14180:Trigonometry
13992:Discriminant
13911:Multivariate
13746:
13544:Ultraproduct
13391:Model theory
13356:Independence
13292:Formal proof
13284:Proof theory
13267:
13240:
13197:real numbers
13169:second-order
13080:Substitution
12957:Metalanguage
12898:conservative
12871:Axiom schema
12815:Constructive
12785:Morse–Kelley
12751:Set theories
12730:Aleph number
12723:inaccessible
12629:Grothendieck
12513:intersection
12400:Higher-order
12388:Second-order
12334:Truth tables
12291:Venn diagram
12074:Formal proof
11810:Key concepts
11737:sieve theory
11633:
11600:
11596:
11592:
11570:
11552:
11528:
11519:Hardy, G. H.
11498:
11471:
11453:
11431:
11413:
11409:
11387:
11356:
11354:
11340:
11332:
11324:
11312:
11300:. Retrieved
11286:
11262:
11258:
11248:
11239:
11230:
11224:
11213:
11204:
11196:
11188:
11180:
11172:
11163:
11155:
11151:
11108:
11100:
11096:
11092:
11084:
11057:
11048:
11017:
11008:
10977:
10953:
10948:
10935:
10923:
10915:
10911:
10906:
10899:
10895:
10890:
10881:
10872:
10860:
10844:
10832:
10823:
10818:for examples
10810:
10802:
10797:
10788:
10779:
10770:
10765:, p. 50
10757:
10748:
10727:
10718:
10685:
10676:
10667:
8539:
8457:
7249:, 7, 9, 11,
6864:, 7, 9, 13,
6290:
6284:
6279:
6273:
6268:
6262:
6233:
6229:
6213:
6185:
6176:
6174:
6164:
6160:
6156:
6146:
6137:
6133:
6129:
6125:
6121:
6117:
6113:
6109:
6105:
6101:
6097:
6093:
6089:
6081:
6077:
6073:
6069:
6065:
6061:
6056:
6044:
6026:such as the
6020:hard problem
6015:
6013:
6010:Cryptography
6005:
5996:
5994:
5978:
5975:Paley graphs
5973:
5970:Graph theory
5958:
5943:
5941:
5921:
5917:
5915:
5905:
5899:
5895:
5891:
5887:
5883:
5879:
5873:
5869:
5865:
5859:
5857:
5851:
5845:
5840:
5834:
5827:
5823:
5767:
5763:
5759:
5752:
5748:
5740:
5737:
5732:
5728:
5726:
5721:
5709:
5705:
5703:
5690:
5686:
5681:
5677:
5671:
5667:
5662:
5658:
5656:
5650:
5646:
5642:
5637:
5633:
5626:
5622:
5618:
5609:
5607:
5601:
5540:
5536:
5489:
5487:
5429:
5382:
5322:
5269:
5267:
5262:
5260:
5246:
5242:
5238:
5234:
5227:
5225:
5181:
5177:
5173:
5161:
5159:
4943:
4937:
4849:
4754:
4752:
4704:
4694:
4692:
4687:
4680:
4674:
4667:
4660:
4656:
4654:
4649:
4645:
4641:
4637:
4633:
4629:
4625:
4621:
4617:
4613:
4609:
4607:
4598:
4594:
4590:
4586:
4578:
4574:
4572:
4562:
4558:
4554:
4550:
4546:
4542:
4531:) (sequence
4528:
4524:
4520:
4516:
4512:
4509:
4505:
4503:
4478:
4474:
4472:
4467:
4463:
4459:
4455:
4450:
4445:
4441:
4437:
4431:
4415:
4411:
4407:
4405:
4400:
4394:
4389:
4384:
4380:
4376:
4374:
4364:
4362:
4241:
4130:
4000:
3861:
3755:
3686:
3682:
3680:
3617:
3502:
3498:
3494:
3490:
3470:
3433:
3423:
3419:
3415:
3411:
3407:
3403:
3401:
3389:
3380:
3371:
3362:
3356:
3352:
3348:
3344:
3340:
3336:
3332:
3328:
3324:
3319:
3308:
3299:
3290:
3281:
3276:
3272:
3268:
3264:
3260:
3256:
3252:
3248:
3243:
3238:
3236:
3118:≡ 3 (mod 4)
3115:
3113:
2999:≡ 1 (mod 4)
2996:
2994:
2961:
2896:
2709:
2705:
2701:
2697:
2693:
2689:
2685:
2681:
2677:
2673:
2669:
2665:
2661:
2657:
2655:
2645:≡ 1 (mod 3)
2642:
2633:
2622:
2613:
2602:
2593:
2585:≡ 1 (mod 4)
2582:
2573:
2561:
2552:
2541:
2532:
2521:
2512:
2501:
2492:
2484:≡ 1 (mod 4)
2481:
2472:
2463:≡ 1 (mod 3)
2460:
2451:
2440:
2431:
2423:≡ 1 (mod 4)
2420:
2411:
2399:
2395:
2390:
2384:
2380:
2375:
2366:
2362:
2358:
2356:
2314:
2197:
2192:
2188:
2184:
2180:
2176:
2172:
2170:
2165:
2161:
2159:
2145:
2141:
2135:
2133:
1984:
1982:
1933:
1929:
1925:
1921:
1919:
1907:
1903:
1899:
1895:
1891:
1887:
1882:
1876:
1872:
1870:
1754:
1752:
1653:
1649:
1642:class number
1631:
1620:
1616:
1612:
1610:
1604:
1600:
1596:
1592:
1574:
1570:cryptography
1562:applications
1557:
1555:
1450:
1446:
1440:
1436:
1345:
1341:
1335:
1331:
1269:
1262:
1193:homomorphism
1186:
994:
962:
954:
949:
945:
941:
937:
933:
925:
923:
918:
913:, which has
871:
808:
804:
799:
795:
791:
787:
783:
779:
777:
773:
769:
764:
760:
756:
752:
750:
746:
740:
737:at least one
736:
732:
728:
724:
720:
713:
709:
705:
701:
697:
693:
687:
675:
670:
666:
662:
658:
654:
650:
646:
644:
641:
633:
632:is even and
629:
625:
621:
617:is a residue
614:
613:is even and
610:
606:
602:
595:
591:
587:
581:
577:
573:
567:
564:
556:
554:
549:
545:
541:
537:
535:
530:
528:
513:
496:
490:
486:
482:
478:
474:
472:
462:
458:
456:
451:
447:
445:
440:
436:
432:
428:
422:
419:
416:
413:
408:
352:
348:
344:
342:prime number
339:
336:
328:
323:
319:
315:
311:
307:
303:
279:
256:
252:
246:
241:
237:
233:
229:
225:
221:
217:
213:
209:
205:
201:
195:
190:
186:
185:For a given
184:
177:
156:
143:cryptography
131:mathematical
128:
123:
119:
115:
113:
53:
49:
37:
31:
30:is called a
27:
18:
14748:Ring theory
14710:Topic lists
14670:Multivector
14656:Free object
14574:Dot product
14560:Determinant
14546:Linear and
13941:Homogeneous
13936:Square-free
13931:Irreducible
13796:Polynomials
13654:Type theory
13602:undecidable
13534:Truth value
13421:equivalence
13100:non-logical
12713:Enumeration
12703:Isomorphism
12650:cardinality
12634:Von Neumann
12599:Ultrafilter
12564:Uncountable
12498:equivalence
12415:Quantifiers
12405:Fixed-point
12374:First-order
12254:Consistency
12239:Proposition
12216:Traditional
12187:Lindström's
12177:Compactness
12119:Type theory
12064:Cardinality
11996:Wikiversity
11918:L-functions
11291:Walker, R.
10867:, p. 9
7460:, 1, 4, 9,
6402:, 1, 4, 7,
5848:≡ 1 (mod 8)
5837:≡ 1 (mod 4)
5714:NP-complete
5232:prime power
3287:= {1,8,15},
3259:, 5, 6, 7,
1902:, 6, 7, 8,
1252:allows its
963:Gauss used
427:is that if
347:there are (
114:Otherwise,
56:such that:
14781:Categories
14726:Glossaries
14680:Polynomial
14660:Free group
14585:Linear map
14442:Inequality
14164:Riemannian
14159:Projective
14144:Symplectic
14139:Hyperbolic
14072:Euclidean
13901:Univariate
13465:elementary
13158:arithmetic
13026:Quantifier
13004:functional
12876:Expression
12594:Transitive
12538:identities
12523:complement
12456:hereditary
12439:Set theory
11877:Arithmetic
11648:PlanetMath
11351:References
11302:25 October
11076:1074.11001
11040:0814.11001
11000:1226.11099
10805:functions.
8430:, 61, 64,
8422:, 46, 49,
8414:, 31, 34,
8402:, 16, 19,
8377:, 39, 41,
8369:, 29, 31,
8357:, 19, 21,
8317:, 65, 67,
8305:, 49, 53,
8261:, 21, 25,
7895:, 31, 34,
7891:, 16, 19,
7848:, 49, 52,
7820:, 13, 16,
7732:, 21, 25,
7488:, 49, 51,
7464:, 14, 16,
7301:, 22, 25,
7201:, 33, 35,
6927:, 1, 4, 7
6776:, 25, 28,
6734:, 1, 2, 4
6226:red number
6198:, and the
5160:For prime
4462:such that
4458:less than
4002:Montgomery
3622:. Setting
3483:Vinogradov
3320:Modulo 19
3244:Modulo 17
2668:+ 1 where
1644:of binary
1587:, and the
318:even) or (
212:) implies
14452:Operation
14185:Lie group
14149:Spherical
13987:Resultant
13926:Trinomial
13906:Bivariate
13736:Supertask
13639:Recursion
13597:decidable
13431:saturated
13409:of models
13332:deductive
13327:axiomatic
13247:Hilbert's
13234:Euclidean
13215:canonical
13138:axiomatic
13070:Signature
12999:Predicate
12888:Extension
12810:Ackermann
12735:Operation
12614:Universal
12604:Recursive
12579:Singleton
12574:Inhabited
12559:Countable
12549:Types of
12533:power set
12503:partition
12420:Predicate
12366:Predicate
12281:Syllogism
12271:Soundness
12244:Inference
12234:Tautology
12136:paradoxes
11635:MathWorld
11591:(1978), "
11375:Gauss sum
8349:, 9, 11,
8321:, 71, 73
8061:, 39, 41
7915:, 9, 11,
7724:, 9, 13,
6751:, 17, 25
6382:, 43, 49
6337:, 23, 25
5987:symmetric
5955:Acoustics
5665:(denoted
5636:≠ ±
5366:−
5306:−
5088:−
5043:±
4971:±
4960:≡
4873:±
4870:≡
4636:≡ 0 (mod
4632:, one if
4616:has 1 + (
4444:) ≪ (log
4345:
4339:
4273:∑
4217:π
4162:∑
4105:
4099:
4061:χ
4029:∑
3964:
3943:π
3879:∑
3862:In fact,
3841:
3773:∑
3732:
3633:χ
3592:
3554:χ
3522:∑
3493:) modulo
3475:coin flip
3410:≡ 2 (mod
3209:−
3191:α
3178:α
3165:α
3130:α
3090:−
3072:α
3059:α
3046:α
3029:−
3011:α
2962:That is,
2909:α
2861:−
2826:∧
2810:−
2774:−
2765:…
2747:∈
2289:−
2280:⋅
2268:−
2251:−
2198:That is:
2085:−
2065:∑
2027:−
2015:π
1956:−
1817:−
1799:∑
1788:π
1783:−
1730:−
1702:∞
1687:∑
1566:acoustics
1339:, and if
1311:−
1258:semigroup
1195:from the
1146:
1127:−
1101:
959:Notations
790:, 7, 8,
536:A number
435:, and if
290:⌋
267:⌊
249:− 1
198:− 1
77:≡
42:congruent
40:if it is
14766:Category
14476:Variable
14466:Relation
14456:Addition
14432:Function
14418:Equation
14367:K-theory
14246:Category
14134:Elliptic
14126:geometry
14107:Polyform
14092:Discrete
14074:geometry
14055:Timeline
14039:Geometry
13972:Division
13921:Binomial
13916:Monomial
13721:Logicism
13714:timeline
13690:Concrete
13549:Validity
13519:T-schema
13512:Kripke's
13507:Tarski's
13502:semantic
13492:Strength
13441:submodel
13436:spectrum
13404:function
13252:Tarski's
13241:Elements
13228:geometry
13184:Robinson
13105:variable
13090:function
13063:spectrum
13053:Sentence
13009:variable
12952:Language
12905:Relation
12866:Automata
12856:Alphabet
12840:language
12694:-jection
12672:codomain
12658:Function
12619:Universe
12589:Infinite
12493:Relation
12276:Validity
12266:Argument
12164:theorem,
11986:Wikibook
11956:Category
11525:(1980),
11495:(1979),
11392:Springer
11183:) steps.
11016:(1994).
10976:(2010).
10623:See also
8394:, 1, 4,
8249:, 7, 9,
8245:, 1, 3,
8085:, 5, 9,
8081:, 1, 3,
7989:, 1, 4,
7883:, 1, 4,
7775:, 5, 9,
7408:, 1, 4,
7340:, 9, 11
7289:, 1, 4,
7165:, 5, 7,
6760:, 1, 4,
6628:, 1, 4,
6346:, 1, 4,
6309:, 1, 3,
6068:) where
6030:and the
4940:Legendre
4854:Lagrange
4652:) = 1.)
4648:and gcd(
4601:) exists
4589:solving
4541:). For
4367:> 0.
3327:, 2, 3,
2995:Then if
2897:and let
1640:for the
863:subgroup
782:, 2, 3,
580:≥
322:+ 1)/2 (
314:/2 + 1 (
200:. Since
170:Legendre
166:Lagrange
145:and the
14678: (
14668: (
14534: (
14524: (
14514: (
14504: (
14494: (
14304:History
14299:Outline
14288:Algebra
14102:Polygon
14050:History
13663:Related
13460:Diagram
13358: (
13337:Hilbert
13322:Systems
13317:Theorem
13195:of the
13140:systems
12920:Formula
12915:Grammar
12831: (
12775:General
12488:Forcing
12473:Element
12393:Monadic
12168:paradox
12109:Theorem
12045:General
11817:Numbers
11367:English
11335:) steps
11279:2690536
11158:) steps
10894:Gauss,
10541:mod 25
10461:mod 24
10381:mod 23
10301:mod 22
10221:mod 21
10141:mod 20
10061:mod 19
9981:mod 18
9901:mod 17
9821:mod 16
9741:mod 15
9661:mod 14
9581:mod 13
9501:mod 12
9421:mod 11
9341:mod 10
8450:A343720
8446:A048152
6575:, 1, 4
6253:A046071
6251::
6243:A096103
6241::
6220:in the
6217:A096008
5934:in the
5931:A000224
5757:. Then
5170:Cipolla
5166:Tonelli
4803:or the
4795:can be
4557:. For
4537:in the
4534:A178153
4491:in the
4488:A053760
4420:√
4392:√
4006:Vaughan
3114:and if
2350:is the
1393:but if
1202:to the
817:of the
727:, then
700:, then
363:of the
240:) (mod
236:−
122:modulo
48:modulo
25:integer
14692:, ...)
14662:, ...)
14589:Matrix
14536:Vector
14526:theory
14516:theory
14512:Module
14506:theory
14496:theory
14335:Scheme
14154:Affine
14087:Convex
13809:degree
13426:finite
13189:Skolem
13142:
13117:Theory
13085:Symbol
13075:String
13058:atomic
12935:ground
12930:closed
12925:atomic
12881:ground
12844:syntax
12740:binary
12667:domain
12584:Finite
12349:finite
12207:Logics
12166:
12114:Theory
11692:Fields
11577:
11559:
11541:
11507:
11478:
11460:
11442:
11420:
11398:
11371:German
11277:
11074:
11064:
11038:
11028:
10998:
10988:
10853:Jacobi
10761:e.g.,
9261:mod 9
9181:mod 8
9101:mod 7
9021:mod 6
8941:mod 5
8861:mod 4
8781:mod 3
8701:mod 2
8621:mod 1
8313:, 63,
8301:, 47,
8293:, 41,
8273:, 33,
8265:, 27,
8253:, 11,
8185:, 25,
8146:, 49,
8134:, 25,
8097:, 15,
8057:, 35,
8049:, 25,
8033:, 13,
8001:, 16,
7963:, 51,
7947:, 39,
7935:, 29,
7832:, 31,
7828:, 25,
7787:, 25,
7744:, 53,
7740:, 49,
7736:, 33,
7695:, 13,
7660:, 37,
7648:, 25,
7581:, 49,
7569:, 37,
7557:, 31,
7553:, 25,
7480:, 36,
7472:, 29,
7357:, 10,
7297:, 16,
7277:, 35,
7265:, 25,
7261:, 23,
7253:, 17,
7217:, 51,
7209:, 39,
7189:, 25,
7185:, 19,
7109:, 25,
7105:, 13,
7029:, 29,
7021:, 16,
6912:, 53,
6896:, 35,
6892:, 33,
6880:, 25,
6876:, 23,
6843:, 25,
6839:, 16,
6796:, 49,
6792:, 43,
6768:, 16,
6704:, 25,
6640:, 16,
6636:, 14,
6608:, 19,
6561:, 49,
6557:, 43,
6553:, 37,
6545:, 31,
6537:, 25,
6533:, 19,
6529:, 13,
6362:, 25,
6358:, 19,
6354:, 16,
6350:, 13,
6333:, 17,
6236:, see
6194:, the
6190:, the
6034:. The
5920:, for
5745:, and
5743:> 1
5249:using
4452:Linnik
3347:, 10,
3271:, 14,
2987:α
2980:α
2973:α
2966:α
2696:+ 1 R
2680:+ 1 R
2315:where
1648:. Let
1583:, the
1575:Using
1542:4 R 15
1538:2 N 15
1536:, but
1274:, the
1254:domain
867:cosets
598:is odd
533:+ 1).
326:odd).
158:Fermat
35:modulo
14630:Trace
14555:Basis
14502:Group
14492:Field
14313:Areas
14216:Lists
14211:Shape
14204:Lists
14173:Other
14062:Lists
13416:Model
13164:Peano
13021:Proof
12861:Arity
12790:Naive
12677:image
12609:Fuzzy
12569:Empty
12518:union
12463:Class
12104:Model
12094:Lemma
12052:Axiom
11832:Unity
11412:[
11365:into
11296:(PDF)
11275:JSTOR
11255:(PDF)
10660:Notes
8385:, 49
8341:, 1,
8212:, 1,
8173:, 1,
8122:, 1,
8029:, 9,
8017:, 3,
8013:, 1,
7911:, 1,
7871:, 9,
7863:, 1,
7808:, 4,
7804:, 1,
7795:, 37
7771:, 1,
7716:, 1,
7687:, 7,
7683:, 1,
7616:, 1,
7525:, 1,
7503:, 1,
7432:, 9,
7428:, 1,
7389:, 9,
7385:, 1,
7353:, 4,
7349:, 1,
7324:, 1,
7313:, 58
7245:, 5,
7241:, 1,
7221:, 59
7169:, 9,
7157:, 1,
7132:, 1,
7097:, 1,
7077:, 49
7041:, 1,
6992:, 1,
6972:, 33
6936:, 1,
6916:, 57
6860:, 5,
6856:, 1,
6847:, 31
6827:, 4,
6823:, 1,
6811:, 1,
6800:, 55
6764:, 7,
6747:, 9,
6743:, 1,
6700:, 9,
6692:, 1,
6667:, 1,
6656:, 49
6632:, 9,
6584:, 1,
6521:, 7,
6517:, 1,
6479:, 25
6471:, 9,
6463:, 1,
6443:, 49
6419:, 9,
6415:, 1,
6313:, 9,
5894:(mod
5766:(mod
5755:) = 1
5625:(mod
5428:, or
5230:is a
4848:, if
4597:(mod
4481:are:
4244:Paley
4133:Schur
3986:0.61.
3479:Pólya
3359:, 18
3343:, 8,
3255:, 3,
2684:, or
1890:, 2,
1544:. If
1349:then
1329:then
1265:cubic
989:3 N 8
985:1 R 8
983:, or
981:5 N 7
977:2 R 7
811:group
786:, 5,
755:, 2,
710:every
628:<
609:<
594:<
563:then
365:field
220:(mod
208:(mod
174:Gauss
162:Euler
44:to a
23:, an
14625:Rank
14605:Norm
14522:Ring
13798:and
13539:Type
13342:list
13146:list
13123:list
13112:Term
13046:rank
12940:open
12834:list
12646:Maps
12551:sets
12410:Free
12380:list
12130:list
12057:list
11575:ISBN
11557:ISBN
11539:ISBN
11505:ISBN
11476:ISBN
11458:ISBN
11440:ISBN
11418:ISBN
11396:ISBN
11369:and
11355:The
11304:2016
11154:log
11062:ISBN
11026:ISBN
10986:ISBN
10814:See
8616:625
7507:, 9
7004:, 9
6499:, 1
6454:, 1
6393:, 1
6249:OEIS
6239:OEIS
6222:OEIS
6147:The
6045:The
5936:OEIS
5872:) =
5864:gcd(
5776:The
5747:gcd(
5738:Let
5699:hard
5631:and
5535:and
4938:and
4539:OEIS
4504:Let
4493:OEIS
4448:).
4414:) ≪
4403:).
4399:log
4316:>
4242:and
4205:>
4004:and
3973:0.41
3934:<
3828:<
3501:and
3481:and
3239:bold
2692:and
2676:and
2639:−12
2619:−11
2599:−10
2191:and
2164:and
2116:>
1908:bold
1853:>
1568:and
1540:and
1495:and
987:and
979:and
967:and
819:ring
800:bold
765:bold
735:for
708:for
13807:By
13226:of
13208:of
13156:of
12688:Sur
12662:Map
12469:Ur-
12451:Set
11646:at
11605:doi
11267:doi
11072:Zbl
11036:Zbl
10996:Zbl
10296:16
10136:17
10056:13
9976:13
9816:10
8613:576
8610:529
8607:484
8604:441
8601:400
8598:361
8595:324
8592:289
8589:256
8586:225
8583:196
8580:169
8577:144
8574:121
8571:100
8568:81
8565:64
8562:49
8559:36
8556:25
8553:16
8534:25
8388:75
8335:50
8326:25
8239:74
8230:49
8206:24
8195:73
8167:48
8158:23
8116:72
8107:47
8075:22
8064:71
8007:46
7983:21
7905:70
7877:45
7857:20
7798:69
7765:44
7756:19
7710:68
7701:43
7677:18
7666:67
7610:42
7601:17
7519:66
7510:41
7497:16
7454:65
7422:40
7402:15
7379:64
7343:39
7318:14
7283:63
7235:38
7226:13
7151:62
7142:37
7126:12
7115:61
7091:36
7082:11
7035:60
7007:35
6986:10
6975:59
6930:34
6850:58
6817:33
6754:57
6737:32
6686:56
6677:31
6622:55
6578:30
6511:54
6502:29
6482:53
6457:28
6409:52
6396:27
6340:51
6303:26
6256:.)
6144:".
5850:if
5839:if
5826:of
5675:or
5600:of
5268:If
5112:mod
5010:mod
4912:mod
4650:a,p
4527:/2,
4429:.
4342:log
4336:log
4258:max
4147:max
4102:log
4096:log
3961:log
3838:log
3729:log
3689:is
3618:in
3589:log
2630:12
2610:11
2590:10
2579:−9
2558:−8
2538:−7
2518:−6
2498:−5
2478:−4
2457:−3
2437:−2
2417:−1
2160:If
1924:if
1660:as
1564:as
1444:or
1014:as
952:.)
936:. (
802:).
767:).
719:if
692:if
624:if
605:if
590:if
576:if
457:If
446:If
282:/2
176:'s
141:to
88:mod
19:In
14783::
14458:,
14424:,
13612:NP
13236::
13230::
13160::
12837:),
12692:Bi
12684:In
11788:,
11764:,
11735:,
11731:,
11717:,
11713:,
11709:,
11705:,
11632:.
11601:16
11599:,
11593:NP
11537:,
11521:;
11491:;
11438:,
11394:,
11273:,
11263:69
11261:,
11257:,
11070:.
11034:.
11020:.
10994:.
10980:.
10972:;
10961:^
10736:^
10706:^
10694:^
10616:0
10604:16
10598:11
10595:24
10592:14
10583:21
10580:19
10577:19
10574:21
10565:14
10562:24
10559:11
10553:16
10536:1
10533:0
10521:16
10515:12
10509:16
10497:0
10485:16
10479:12
10473:16
10456:4
10450:0
10438:16
10432:13
10426:18
10423:12
10408:12
10405:18
10399:13
10393:16
10376:9
10367:0
10355:16
10349:14
10343:20
10340:15
10337:12
10334:11
10331:12
10328:15
10325:20
10319:14
10313:16
10284:0
10272:16
10266:15
10257:18
10254:16
10251:16
10248:18
10239:15
10233:16
10216:5
10213:16
10201:0
10189:16
10183:16
10171:0
10159:16
10153:16
10130:16
10118:0
10106:16
10100:17
10097:11
10082:11
10079:17
10073:16
10047:16
10035:0
10023:16
10014:13
10011:10
10005:10
10002:13
9993:16
9973:15
9964:16
9952:0
9940:16
9931:15
9928:13
9925:13
9922:15
9913:16
9896:1
9893:0
9869:0
9845:0
9801:10
9786:0
9771:10
9756:10
9736:9
9730:11
9718:11
9703:0
9688:11
9676:11
9656:1
9644:12
9641:10
9638:10
9635:12
9620:0
9605:12
9602:10
9599:10
9596:12
9576:1
9573:0
9555:0
9537:0
9519:0
9496:9
9487:0
9454:0
9416:5
9401:0
9371:0
9336:4
9315:0
9288:0
9256:1
9253:0
9241:0
9229:0
9217:0
9205:0
9193:0
9176:2
9164:0
9143:0
9122:0
9096:1
9093:0
9075:0
9057:0
9039:0
9016:0
9001:0
8986:0
8971:0
8956:0
8936:1
8933:0
8927:0
8921:0
8915:0
8909:0
8903:0
8897:0
8891:0
8885:0
8879:0
8873:0
8867:0
8856:1
8853:0
8844:0
8835:0
8826:0
8817:0
8808:0
8799:0
8790:0
8776:1
8773:0
8767:0
8761:0
8755:0
8749:0
8743:0
8737:0
8731:0
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5807:=
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5797:p
5794:a
5788:(
5770:)
5768:n
5764:a
5760:x
5753:n
5751:,
5749:a
5741:n
5733:n
5729:a
5722:c
5710:c
5706:x
5691:n
5687:n
5682:n
5678:a
5672:n
5668:a
5663:n
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5651:n
5647:n
5643:n
5638:y
5634:x
5629:)
5627:n
5623:y
5619:x
5610:n
5602:n
5584:1
5581:=
5577:)
5570:2
5566:/
5562:n
5558:a
5552:(
5541:n
5537:n
5523:1
5520:=
5516:)
5510:n
5507:a
5501:(
5490:n
5473:1
5470:=
5466:)
5459:2
5455:/
5451:n
5447:a
5441:(
5430:n
5416:1
5413:=
5409:)
5403:n
5400:a
5394:(
5383:n
5369:1
5363:=
5359:)
5352:2
5348:/
5344:n
5340:a
5334:(
5323:n
5309:1
5303:=
5299:)
5293:n
5290:a
5284:(
5270:n
5263:n
5247:n
5243:p
5239:p
5235:n
5228:n
5206:)
5201:n
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5193:(
5182:x
5178:n
5174:n
5162:n
5138:n
5130:a
5119:)
5116:n
5109:(
5102:4
5098:/
5094:)
5091:1
5085:n
5082:(
5078:2
5072:8
5068:/
5064:)
5061:3
5058:+
5055:n
5052:(
5048:a
5036:n
5028:a
5017:)
5014:n
5007:(
5000:8
4996:/
4992:)
4989:3
4986:+
4983:n
4980:(
4976:a
4965:{
4957:x
4944:n
4923:,
4919:)
4916:n
4909:(
4902:4
4898:/
4894:)
4891:1
4888:+
4885:n
4882:(
4878:a
4867:x
4850:n
4836:1
4833:=
4829:)
4824:n
4821:a
4816:(
4782:)
4777:n
4774:a
4769:(
4755:n
4695:n
4688:n
4684:2
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4638:p
4634:a
4630:p
4626:a
4622:p
4620:|
4618:a
4614:a
4610:p
4599:n
4595:a
4591:x
4587:x
4579:n
4575:a
4563:E
4559:p
4555:r
4553:−
4551:p
4547:r
4543:p
4529:p
4525:p
4521:p
4517:p
4515:(
4513:E
4506:p
4495:)
4479:p
4475:p
4468:p
4466:(
4464:n
4460:X
4456:p
4446:p
4442:p
4440:(
4438:n
4422:e
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4412:p
4410:(
4408:n
4401:p
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4385:p
4383:(
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4365:d
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4331:d
4324:7
4321:1
4312:|
4307:)
4302:n
4299:d
4294:(
4288:N
4283:1
4280:=
4277:n
4268:|
4262:N
4225:q
4214:2
4210:1
4201:|
4196:)
4191:q
4188:n
4183:(
4177:N
4172:1
4169:=
4166:n
4157:|
4151:N
4116:.
4112:)
4108:q
4091:q
4085:(
4081:O
4078:=
4074:|
4070:)
4067:n
4064:(
4056:N
4053:+
4050:M
4045:1
4042:+
4039:M
4036:=
4033:n
4024:|
3983:+
3978:q
3970:+
3967:q
3956:q
3947:2
3939:4
3930:|
3925:)
3920:q
3917:n
3912:(
3906:N
3903:+
3900:M
3895:1
3892:+
3889:M
3886:=
3883:n
3874:|
3847:.
3844:q
3833:q
3824:|
3819:)
3814:q
3811:n
3806:(
3800:N
3797:+
3794:M
3789:1
3786:+
3783:M
3780:=
3777:n
3768:|
3741:.
3738:)
3735:q
3724:q
3719:(
3716:O
3713:+
3710:N
3705:2
3702:1
3687:N
3683:q
3666:,
3662:)
3657:q
3654:n
3649:(
3645:=
3642:)
3639:n
3636:(
3603:,
3599:)
3595:q
3584:q
3578:(
3574:O
3571:=
3567:|
3563:)
3560:n
3557:(
3549:N
3546:+
3543:M
3538:1
3535:+
3532:M
3529:=
3526:n
3517:|
3503:N
3499:M
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3491:n
3471:a
3457:)
3451:p
3448:a
3442:(
3424:b
3420:a
3416:p
3412:p
3408:x
3404:p
3390:A
3381:A
3372:A
3363:A
3345:9
3341:7
3337:6
3333:5
3329:4
3325:1
3309:A
3300:A
3291:A
3282:A
3265:9
3261:8
3257:4
3253:2
3249:1
3221:.
3216:4
3212:3
3206:p
3200:=
3187:=
3174:=
3160:,
3155:4
3151:1
3148:+
3145:p
3139:=
3116:p
3097:4
3093:1
3087:p
3081:=
3068:=
3055:=
3041:,
3036:4
3032:5
3026:p
3020:=
2997:p
2947:.
2943:|
2937:j
2934:i
2930:A
2925:|
2921:=
2916:j
2913:i
2882:,
2878:}
2872:j
2868:)
2864:1
2858:(
2855:=
2851:)
2846:p
2842:1
2839:+
2836:k
2830:(
2821:i
2817:)
2813:1
2807:(
2804:=
2800:)
2795:p
2792:k
2787:(
2783::
2780:}
2777:2
2771:p
2768:,
2762:,
2759:2
2756:,
2753:1
2750:{
2744:k
2740:{
2736:=
2731:j
2728:i
2724:A
2710:j
2706:i
2702:p
2698:p
2694:n
2690:p
2686:n
2682:p
2678:n
2674:p
2670:n
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2658:p
2643:p
2634:p
2623:p
2614:p
2603:p
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2583:p
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2533:p
2522:p
2513:p
2502:p
2493:p
2482:p
2473:p
2461:p
2452:p
2441:p
2432:p
2421:p
2412:p
2400:p
2396:a
2391:a
2385:p
2381:a
2376:a
2367:a
2363:p
2359:a
2337:)
2332:q
2329:p
2324:(
2296:2
2292:1
2286:q
2275:2
2271:1
2265:p
2258:)
2254:1
2248:(
2245:=
2241:)
2236:p
2233:q
2228:(
2223:)
2218:q
2215:p
2210:(
2193:q
2189:p
2185:p
2181:q
2177:q
2173:p
2166:q
2162:p
2136:q
2112:)
2107:q
2104:n
2099:(
2092:2
2088:1
2082:q
2075:1
2072:=
2069:n
2056:q
2049:)
2044:)
2039:q
2036:2
2031:(
2024:2
2020:(
2010:=
2007:)
2004:1
2001:(
1998:L
1985:q
1966:4
1962:)
1959:1
1953:q
1950:(
1947:q
1934:q
1930:q
1926:q
1922:q
1910:)
1904:9
1900:5
1896:4
1892:3
1888:1
1877:q
1873:q
1849:)
1844:q
1841:n
1836:(
1830:q
1827:n
1820:1
1814:q
1809:1
1806:=
1803:n
1792:q
1780:=
1777:)
1774:1
1771:(
1768:L
1755:q
1738:.
1733:s
1726:n
1721:)
1716:q
1713:n
1708:(
1697:1
1694:=
1691:n
1683:=
1680:)
1677:s
1674:(
1671:L
1654:s
1650:q
1621:p
1617:p
1613:p
1605:p
1601:M
1597:p
1593:M
1558:n
1546:m
1524:1
1521:=
1518:)
1509:4
1503:(
1483:1
1480:=
1477:)
1468:2
1462:(
1451:m
1447:a
1441:m
1437:a
1422:1
1419:=
1416:)
1410:m
1407:a
1401:(
1381:,
1378:1
1375:=
1372:)
1366:m
1363:a
1357:(
1346:m
1342:a
1336:m
1332:a
1317:,
1314:1
1308:=
1305:)
1299:m
1296:a
1290:(
1280:m
1272:p
1240:0
1237:=
1234:)
1228:p
1224:p
1221:n
1214:(
1199:p
1189:p
1165:a
1157:p
1149:p
1143:N
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1130:1
1120:a
1112:p
1104:p
1098:R
1095:a
1085:1
1082:+
1075:a
1067:p
1057:0
1048:{
1043:=
1039:)
1034:p
1031:a
1026:(
1012:p
1005:a
991:.
969:N
965:R
950:n
946:a
942:n
938:a
934:n
926:a
919:n
901:)
897:Z
893:n
889:/
884:Z
880:(
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845:Z
841:n
837:/
832:Z
828:(
792:9
788:6
784:4
780:1
761:4
757:3
753:1
743:.
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729:a
725:n
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716:.
714:n
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702:a
698:n
694:a
671:n
667:p
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659:p
655:p
651:p
647:n
634:a
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622:p
615:a
611:n
607:k
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596:n
592:k
588:p
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578:k
574:p
568:a
565:p
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491:a
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475:a
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429:p
409:p
395:)
391:Z
387:p
383:/
378:Z
374:(
353:p
349:p
345:p
324:n
320:n
316:n
312:n
308:n
304:n
280:n
257:n
253:n
247:n
242:n
238:a
234:n
230:a
226:n
222:n
218:b
216:≡
214:a
210:n
206:b
204:≡
202:a
196:n
191:n
187:n
124:n
116:q
99:.
95:)
92:n
85:(
80:q
72:2
68:x
54:x
50:n
38:n
28:q
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