1749:
4861:
1046:
1744:{\displaystyle {\begin{alignedat}{2}a\cdot b&=2^{a_{1}+b_{1}}3^{a_{2}+b_{2}}5^{a_{3}+b_{3}}7^{a_{4}+b_{4}}\cdots &&=\prod p_{i}^{a_{i}+b_{i}},\\\gcd(a,b)&=2^{\min(a_{1},b_{1})}3^{\min(a_{2},b_{2})}5^{\min(a_{3},b_{3})}7^{\min(a_{4},b_{4})}\cdots &&=\prod p_{i}^{\min(a_{i},b_{i})},\\\operatorname {lcm} (a,b)&=2^{\max(a_{1},b_{1})}3^{\max(a_{2},b_{2})}5^{\max(a_{3},b_{3})}7^{\max(a_{4},b_{4})}\cdots &&=\prod p_{i}^{\max(a_{i},b_{i})}.\end{alignedat}}}
42:
3308:
4337:
The famous physicist and mathematician Kamal al-Din al-Farisi compiled a paper in which he set out deliberately to prove the theorem of Ibn Qurra in an algebraic way. This forced him to an understanding of the first arithmetical functions and to a full preparation which led him to state for the first
4165:
Any commutative Möbius monoid satisfies a unique factorization theorem and thus possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers. Fundamental
Theorem of Arithmetic is, in fact, a special case of the unique factorization theorem in commutative
253:
4440:
has been translated from Latin into
English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
3273:
Therefore, there cannot exist a smallest integer with more than a single distinct prime factorization. Every positive integer must either be a prime number itself, which would factor uniquely, or a composite that also factors uniquely into primes, or in the case of the integer
4283:, p. 5): "Even in Euclid, we fail to find a general statement about the uniqueness of the factorization of an integer into primes; surely he may have been aware of it, but all he has is a statement (Eucl.IX.I4) about the l.c.m. of any number of given primes."
2423:
985:
735:
3750:
540:
of several prime numbers is not a multiple of any other prime number.) Book IX, proposition 14 is derived from Book VII, proposition 30, and proves partially that the decomposition is unique – a point critically noted by
97:
4306:
One could say that Euclid takes the first step on the way to the existence of prime factorization, and al-Farisi takes the final step by actually proving the existence of a finite prime factorization in his first
4488:
The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76. Footnotes referencing these are of the form "Gauss, BQ, §
2971:
2319:
3527:
374:
3616:
4689:
410:. The implicit use of unique factorization in rings of algebraic integers is behind the error of many of the numerous false proofs that have been written during the 358 years between
2733:
311:
3429:, that the non-zero, non-unit numbers fall into two classes, primes and composites, and that (except for order), the composites have unique factorization as a product of primes (
2684:
3951:
3911:
3867:
3793:
2588:
2314:
2788:
17:
2831:
2635:
262:
be represented as a product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product.
4108:
4067:
3976:
3560:
3423:
4033:
4002:
3657:
3473:
3184:
2526:
3807:
must be a unit. This is the traditional definition of "prime". It can also be proven that none of these factors obeys Euclid's lemma; for example, 2 divides neither (1 +
3045:
847:
2860:
3214:
3114:
2483:
599:
3268:
3241:
3141:
3084:
3005:
2453:
3292:
2306:
2282:
2262:
2237:
The fundamental theorem of arithmetic can also be proved without using Euclid's lemma. The proof that follows is inspired by Euclid's original version of the
4320:
3665:
440:
If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.
3325:
522:
If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it.
248:{\displaystyle 1200=2^{4}\cdot 3^{1}\cdot 5^{2}=(2\cdot 2\cdot 2\cdot 2)\cdot 3\cdot (5\cdot 5)=5\cdot 2\cdot 5\cdot 2\cdot 3\cdot 2\cdot 2=\ldots }
1758:, especially of large numbers, is much more difficult than computing products, GCDs, or LCMs. So these formulas have limited use in practice.
415:
4804:
496:(In modern terminology: every integer greater than one is divided evenly by some prime number.) Proposition 31 is proved directly by
4151:, special subsets of rings. Multiplication is defined for ideals, and the rings in which they have unique factorization are called
774:
are positive integers. This representation is commonly extended to all positive integers, including 1, by the convention that the
4713:
2868:
2264:
is the smallest positive integer which is the product of prime numbers in two different ways. Incidentally, this implies that
5212:
4698:
4602:
4561:
4461:
4267:, the factorization into prime elements may be non unique, but one can recover a unique factorization if one factors into
1051:
4472:
Untersuchungen über hohere
Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition)
323:
4528:, Vol II, pp. 65–92 and 93–148; German translations are pp. 511–533 and 534–586 of the German edition of the
3573:
3478:
4678:
4479:
4395:
4330:
3347:
5047:
4453:
553:
426:
The fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of
35:
4860:
4755:
3329:
31:
4797:
5217:
4733:
545:. Indeed, in this proposition the exponents are all equal to one, so nothing is said for the general case.
56:
2689:
2418:{\displaystyle {\begin{aligned}s&=p_{1}p_{2}\cdots p_{m}\\&=q_{1}q_{2}\cdots q_{n}.\end{aligned}}}
272:
4436:
4113:
2643:
565:
383:
47:
5001:
5037:
4264:
3916:
3876:
3832:
3758:
3621:
However, it was also discovered that unique factorization does not always hold. An example is given by
2531:
1772:
Many arithmetic functions are defined using the canonical representation. In particular, the values of
4447:
2738:
5022:
2793:
2597:
497:
4751:
3795:
it can be proven that if any of the factors above can be represented as a product, for example, 2 =
4790:
4741:
4737:
4072:
4038:
3982:
a prime must be irreducible. Euclid's classical lemma can be rephrased as "in the ring of integers
3956:
3532:
3394:
407:
5166:
4728:
4007:
3985:
3624:
3447:
3146:
2488:
980:{\displaystyle n=2^{n_{1}}3^{n_{2}}5^{n_{3}}7^{n_{4}}\cdots =\prod _{i=1}^{\infty }p_{i}^{n_{i}},}
5176:
5042:
4966:
4775:
4632:
4594:
3822:
3361:
3318:
3010:
1777:
1017:
406:. This failure of unique factorization is one of the reasons for the difficulty of the proof of
5222:
5027:
4986:
4129:
730:{\displaystyle n=p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots p_{k}^{n_{k}}=\prod _{i=1}^{k}p_{i}^{n_{i}},}
387:
1990:
Suppose, to the contrary, there is an integer that has two distinct prime factorizations. Let
518:
Proposition 32 is derived from proposition 31, and proves that the decomposition is possible.
4956:
4825:
4175:
2836:
1755:
1021:
537:
3189:
3089:
2458:
317:
5130:
5032:
4612:
3246:
3219:
3119:
3062:
2983:
2431:
560:
432:
52:
4620:
8:
5191:
5186:
4981:
4976:
4961:
4900:
4586:
4549:
4268:
4148:
4121:
3826:
3441:
2238:
1767:
556:
took the final step and stated for the first time the fundamental theorem of arithmetic.
399:
379:
5115:
5110:
5071:
4991:
4971:
4159:
3567:
3360:
The first generalization of the theorem is found in Gauss's second monograph (1832) on
3277:
2291:
2267:
2247:
570:
4782:
2217:. We now have two distinct prime factorizations of some integer strictly smaller than
5151:
5091:
4694:
4674:
4657:
4636:
4598:
4582:
4557:
4542:
4475:
4457:
4391:
4326:
3437:
2122:
1789:
1773:
475:
403:
4112:
The rings in which factorization into irreducibles is essentially unique are called
320:: if 1 were prime, then factorization into primes would not be unique; for example,
5181:
5156:
5076:
5062:
4996:
4880:
4840:
4616:
4144:
4125:
3369:
2285:
1817:
838:
411:
391:
266:
3745:{\displaystyle 6=2\cdot 3=\left(1+{\sqrt {-5}}\right)\left(1-{\sqrt {-5}}\right).}
265:
The requirement that the factors be prime is necessary: factorizations containing
5161:
5086:
5080:
5017:
4915:
4905:
4835:
4718:
4608:
4181:
4152:
4117:
3979:
1800:
the product of two integers, then it must divide at least one of these integers.
1005:
395:
5171:
5125:
4951:
4935:
4925:
4895:
4294:
3373:
4763:
5206:
5120:
4920:
4910:
4890:
4653:
3870:
3563:
775:
3913:(if it divides a product it must divide one of the factors). The mention of
542:
5135:
5052:
4930:
4875:
4845:
4745:
4722:
4590:
4574:
4140:
4136:
478:, and it is the key in the proof of the fundamental theorem of arithmetic.
84:
4771:
4570:
3365:
1780:
functions are determined by their values on the powers of prime numbers.
64:
1836:
is prime, there is nothing more to prove. Otherwise, there are integers
552:
took the first step on the way to the existence of prime factorization,
3332: in this section. Unsourced material may be challenged and removed.
4162:, though it requires some additional conditions to ensure uniqueness.
4885:
4690:
Number Theory: An
Approach through History from Hammurapi to Legendre
4671:
Prime
Numbers and Computer Methods for Factorization (second edition)
2594:
and has two distinct prime factorizations. One may also suppose that
1032:
can be expressed simply in terms of the canonical representations of
41:
3307:
837:). In fact, any positive integer can be uniquely represented as an
3755:
Examples like this caused the notion of "prime" to be modified. In
1004:
Allowing negative exponents provides a canonical form for positive
593:
can be represented in exactly one way as a product of prime powers
55:
proved the unique factorization theorem and used it to prove the
4830:
4640:
4292:
1797:
581:
258:
The theorem says two things about this example: first, that 1200
80:
4661:
4388:
Why Prove it Again? Alternative Proofs in
Mathematical Practice.
504:
Any number either is prime or is measured by some prime number.
4537:
549:
427:
4295:"A Historical Survey of the Fundamental Theorem of Arithmetic"
5101:
4553:
3430:
88:
4184: – Multiset of prime exponents in a prime factorization
83:
greater than 1 can be represented uniquely as a product of
4474:(in German), translated by Maser, H., New York: Chelsea,
2980:
have been supposed to have a unique prime factorization,
4714:
Why isn’t the fundamental theorem of arithmetic obvious?
4449:
2966:{\displaystyle s-p_{1}Q=(q_{1}-p_{1})Q=p_{1}(P-Q)<s.}
4812:
4517:
Theoria residuorum biquadraticorum, Commentatio secunda
482:
Any composite number is measured by some prime number.
4729:
PlanetMath: Proof of fundamental theorem of arithmetic
4693:, Modern Birkhäuser Classics, Boston, MA: Birkhäuser,
3481:
4508:
Theoria residuorum biquadraticorum, Commentatio prima
4075:
4041:
4010:
3988:
3978:
Using these definitions it can be proven that in any
3959:
3919:
3879:
3835:
3761:
3668:
3627:
3576:
3535:
3450:
3397:
3280:
3249:
3222:
3192:
3149:
3122:
3092:
3065:
3013:
2986:
2871:
2839:
2796:
2741:
2692:
2646:
2600:
2534:
2491:
2461:
2434:
2317:
2294:
2270:
2250:
1049:
850:
602:
326:
275:
100:
3425:
He showed that this ring has the four units ±1 and ±
2232:
4648:Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970),
2183:, we may cancel these two factors to conclude that
1820:, assume this is true for all numbers greater than
4541:
4519:, Göttingen: Comment. Soc. regiae sci, Göttingen 7
4510:, Göttingen: Comment. Soc. regiae sci, Göttingen 6
4199:
4197:
4102:
4061:
4027:
4004:every irreducible is prime". This is also true in
3996:
3970:
3953:is required because 2 is prime and irreducible in
3945:
3905:
3861:
3787:
3744:
3651:
3610:
3554:
3522:{\textstyle \omega ={\frac {-1+{\sqrt {-3}}}{2}},}
3521:
3467:
3417:
3286:
3262:
3235:
3208:
3178:
3135:
3108:
3078:
3039:
2999:
2965:
2854:
2825:
2782:
2727:
2678:
2629:
2582:
2520:
2477:
2447:
2417:
2300:
2276:
2256:
1743:
979:
729:
369:{\displaystyle 2=2\cdot 1=2\cdot 1\cdot 1=\ldots }
368:
305:
247:
4647:
4361:
4239:
3611:{\displaystyle \pm 1,\pm \omega ,\pm \omega ^{2}}
2637:by exchanging the two factorizations, if needed.
1808:It must be shown that every integer greater than
1001:are positive integers, and the others are zero.
569:is an early modern statement and proof employing
5204:
4734:Fermat's Last Theorem Blog: Unique Factorization
4552:(Second Edition Unabridged ed.), New York:
4178: – Decomposition of a number into a product
1812:is either prime or a product of primes. First,
1700:
1642:
1603:
1564:
1525:
1451:
1393:
1354:
1315:
1276:
1246:
778:is equal to 1 (the empty product corresponds to
4194:
3364:. This paper introduced what is now called the
3059:, must have a unique prime factorization, and
1016:The canonical representations of the product,
841:taken over all the positive prime numbers, as
827:may be inserted without changing the value of
582:Canonical representation of a positive integer
318:reasons why 1 is not considered a prime number
18:Canonical representation of a positive integer
4798:
4719:GCD and the Fundamental Theorem of Arithmetic
4452:, translated by Clarke, Arthur A., New York:
4322:Encyclopedia of the History of Arabic Science
3869:(only divisible by itself or a unit) but not
3821:) even though it divides their product 6. In
2528:then there would exist some positive integer
4548:, vol. 2 (Books III-IX), Translated by
4569:
4536:
4420:
4373:
4338:time the fundamental theorem of arithmetic.
4251:
3116:The former case is also impossible, as, if
528:
510:
488:
474:or both.) Proposition 30 is referred to as
446:
4805:
4791:
3007:must occur in the factorization of either
4077:
4043:
4012:
3990:
3961:
3921:
3881:
3837:
3763:
3629:
3452:
3399:
3348:Learn how and when to remove this message
402:. However, the theorem does not hold for
27:Integers have unique prime factorizations
4629:Elementary Introduction to Number Theory
4579:An Introduction to the Theory of Numbers
3433:the order and multiplication by units).
1011:
40:
4293:A. Goksel Agargun and E. Mehmet Özkan.
1761:
91:the order of the factors. For example,
14:
5205:
4668:
4385:
4318:
3618:and that it has unique factorization.
2223:, which contradicts the minimality of
4786:
4514:
4505:
4469:
4445:
4409:
4215:
4203:
3570:, and he proved it has the six units
2177:. Returning to our factorizations of
4736:, a blog that covers the history of
4686:
4626:
4349:
4280:
4227:
3436:Similarly, in 1844 while working on
3330:adding citations to reliable sources
3301:
3051:. The latter case is impossible, as
2728:{\displaystyle Q=q_{2}\cdots q_{n},}
1996:be the least such integer and write
454:(In modern terminology: if a prime
306:{\displaystyle 12=2\cdot 6=3\cdot 4}
4813:Divisibility-based sets of integers
4752:"Fundamental Theorem of Arithmetic"
3391:are integers. It is now denoted by
2976:As the positive integers less than
2679:{\displaystyle P=p_{2}\cdots p_{m}}
24:
4544:The thirteen books of the Elements
3297:
2125:. Without loss of generality, say
947:
788:This representation is called the
25:
5234:
4851:Fundamental theorem of arithmetic
4761:
4707:
4497:are of the form "Gauss, DA, Art.
4160:unique factorization for ordinals
4147:(1876) into the modern theory of
4143:, which was developed further by
3946:{\displaystyle \mathbb {Z} \left}
3906:{\displaystyle \mathbb {Z} \left}
3862:{\displaystyle \mathbb {Z} \left}
3788:{\displaystyle \mathbb {Z} \left}
2583:{\displaystyle t=s/p_{i}=s/q_{j}}
2233:Uniqueness without Euclid's lemma
1929:are products of primes. But then
378:The theorem generalizes to other
269:may not be unique (for example,
69:fundamental theorem of arithmetic
4859:
3306:
2783:{\displaystyle s=p_{1}P=q_{1}Q.}
2161:are both prime, it follows that
1873:. By the induction hypothesis,
316:This theorem is one of the main
4778:from the original on 2021-12-11
4414:
4403:
4379:
4367:
4355:
4343:
3317:needs additional citations for
2826:{\displaystyle p_{1}<q_{1},}
2630:{\displaystyle p_{1}<q_{1},}
576:
36:Fundamental theorem of calculus
4756:Wolfram Demonstrations Project
4515:Gauss, Carl Friedrich (1832),
4506:Gauss, Carl Friedrich (1828),
4470:Gauss, Carl Friedrich (1965),
4446:Gauss, Carl Friedrich (1986),
4362:Pettofrezzo & Byrkit (1970
4312:
4286:
4274:
4257:
4245:
4240:Pettofrezzo & Byrkit (1970
4233:
4221:
4209:
4094:
4081:
4053:
4047:
4022:
4016:
3646:
3633:
3462:
3456:
3409:
3403:
2951:
2939:
2920:
2894:
1729:
1703:
1671:
1645:
1632:
1606:
1593:
1567:
1554:
1528:
1510:
1498:
1480:
1454:
1422:
1396:
1383:
1357:
1344:
1318:
1305:
1279:
1261:
1249:
194:
182:
170:
146:
32:Fundamental theorem of algebra
13:
1:
4493:". Footnotes referencing the
4429:
4319:Rashed, Roshdi (2002-09-11).
4103:{\displaystyle \mathbb {Z} .}
4062:{\displaystyle \mathbb {Z} ,}
3971:{\displaystyle \mathbb {Z} .}
3555:{\displaystyle \omega ^{3}=1}
3418:{\displaystyle \mathbb {Z} .}
3294:, not factor into any prime.
3186:it must be also a divisor of
1985:
990:where a finite number of the
5213:Theorems about prime numbers
4120:over the integers or over a
4114:unique factorization domains
4028:{\displaystyle \mathbb {Z} }
3997:{\displaystyle \mathbb {Z} }
3652:{\displaystyle \mathbb {Z} }
3468:{\displaystyle \mathbb {Z} }
3179:{\displaystyle q_{1}-p_{1},}
2521:{\displaystyle p_{i}=q_{j},}
2455:must be distinct from every
1803:
384:unique factorization domains
73:unique factorization theorem
57:law of quadratic reciprocity
7:
4631:(2nd ed.), Lexington:
4495:Disquisitiones Arithmeticae
4437:Disquisitiones Arithmeticae
4169:
3040:{\displaystyle q_{1}-p_{1}}
566:Disquisitiones Arithmeticae
77:prime factorization theorem
48:Disquisitiones Arithmeticae
10:
5239:
4593:. (6th ed.), Oxford:
4325:. Routledge. p. 385.
4265:ring of algebraic integers
4139:introduced the concept of
2284:, if it exists, must be a
1765:
536:(In modern terminology: a
421:
29:
5144:
5100:
5061:
5048:Superior highly composite
5010:
4944:
4868:
4857:
4818:
4650:Elements of Number Theory
4116:. Important examples are
4945:Constrained divisor sums
4742:Diophantus of Alexandria
4627:Long, Calvin T. (1972),
4421:Hardy & Wright (2008
4390:, Springer, p. 45,
4386:Dawson, John W. (2015),
4374:Hardy & Wright (2008
4252:Hardy & Wright (2008
4188:
1982:is a product of primes.
1783:
790:canonical representation
30:Not to be confused with
4633:D. C. Heath and Company
4595:Oxford University Press
4130:principal ideal domains
3823:algebraic number theory
3659:. In this ring one has
3362:biquadratic reciprocity
3216:which is impossible as
2855:{\displaystyle Q<P.}
1018:greatest common divisor
586:Every positive integer
388:principal ideal domains
4673:, Boston: Birkhäuser,
4158:There is a version of
4104:
4063:
4029:
3998:
3972:
3947:
3907:
3863:
3789:
3746:
3653:
3612:
3566:. This is the ring of
3556:
3523:
3469:
3419:
3288:
3264:
3237:
3210:
3209:{\displaystyle q_{1},}
3180:
3137:
3110:
3109:{\displaystyle q_{j}.}
3080:
3041:
3001:
2967:
2856:
2827:
2784:
2729:
2680:
2631:
2584:
2522:
2479:
2478:{\displaystyle q_{j}.}
2449:
2419:
2302:
2278:
2258:
2068:is prime. We see that
1745:
981:
951:
731:
701:
554:Kamāl al-Dīn al-Fārisī
534:
516:
494:
452:
370:
307:
249:
60:
4826:Integer factorization
4764:"1 and Prime Numbers"
4738:Fermat's Last Theorem
4687:Weil, André (2007) ,
4669:Riesel, Hans (1994),
4524:These are in Gauss's
4176:Integer factorization
4105:
4064:
4030:
3999:
3973:
3948:
3908:
3864:
3790:
3747:
3654:
3613:
3557:
3524:
3470:
3420:
3289:
3270:are distinct primes.
3265:
3263:{\displaystyle q_{1}}
3238:
3236:{\displaystyle p_{1}}
3211:
3181:
3138:
3136:{\displaystyle p_{1}}
3111:
3081:
3079:{\displaystyle p_{1}}
3055:, being smaller than
3042:
3002:
3000:{\displaystyle p_{1}}
2968:
2862:It then follows that
2857:
2828:
2785:
2730:
2681:
2632:
2590:that is smaller than
2585:
2523:
2480:
2450:
2448:{\displaystyle p_{i}}
2420:
2303:
2279:
2259:
1796:VII, 30): If a prime
1756:integer factorization
1746:
1024:(LCM) of two numbers
1022:least common multiple
1012:Arithmetic operations
982:
931:
732:
681:
538:least common multiple
520:
502:
480:
438:
408:Fermat's Last Theorem
371:
308:
250:
44:
4652:, Englewood Cliffs:
4302:Historia Mathematica
4073:
4039:
4008:
3986:
3957:
3917:
3877:
3833:
3759:
3666:
3625:
3574:
3533:
3479:
3448:
3444:introduced the ring
3395:
3326:improve this article
3278:
3247:
3220:
3190:
3147:
3120:
3090:
3063:
3011:
2984:
2869:
2837:
2794:
2739:
2690:
2644:
2598:
2532:
2489:
2459:
2432:
2315:
2292:
2268:
2248:
1816:is prime. Then, by
1762:Arithmetic functions
1047:
848:
600:
458:divides the product
380:algebraic structures
324:
273:
98:
79:, states that every
5218:Uniqueness theorems
5038:Colossally abundant
4869:Factorization forms
4550:Thomas Little Heath
4410:Gauss, BQ, §§ 31–34
3568:Eisenstein integers
3086:differs from every
2239:Euclidean algorithm
1768:Arithmetic function
1733:
1484:
1238:
973:
763:are primes and the
723:
677:
652:
630:
5023:Primitive abundant
5011:With many divisors
4100:
4059:
4025:
3994:
3968:
3943:
3903:
3859:
3785:
3742:
3649:
3608:
3552:
3519:
3465:
3415:
3284:
3260:
3233:
3206:
3176:
3133:
3106:
3076:
3037:
2997:
2963:
2852:
2823:
2780:
2725:
2676:
2627:
2580:
2518:
2485:Otherwise, if say
2475:
2445:
2415:
2413:
2298:
2274:
2254:
1741:
1739:
1690:
1441:
1204:
977:
952:
727:
702:
656:
631:
609:
571:modular arithmetic
404:algebraic integers
366:
303:
245:
71:, also called the
61:
5200:
5199:
4754:by Hector Zenil,
4700:978-0-817-64565-6
4604:978-0-19-921986-5
4583:D. R. Heath-Brown
4563:978-0-486-60089-5
4463:978-0-387-96254-2
4126:Euclidean domains
4092:
3937:
3897:
3853:
3814:) nor (1 −
3779:
3732:
3706:
3644:
3514:
3508:
3438:cubic reciprocity
3372:, the set of all
3370:Gaussian integers
3358:
3357:
3350:
3287:{\displaystyle 1}
2301:{\displaystyle 1}
2277:{\displaystyle s}
2257:{\displaystyle s}
511:Elements Book VII
489:Elements Book VII
447:Elements Book VII
392:Euclidean domains
267:composite numbers
16:(Redirected from
5230:
5177:Harmonic divisor
5063:Aliquot sequence
5043:Highly composite
4967:Multiply perfect
4863:
4841:Divisor function
4807:
4800:
4793:
4784:
4783:
4779:
4744:to the proof by
4703:
4683:
4664:
4643:
4623:
4566:
4547:
4520:
4511:
4484:
4466:
4424:
4418:
4412:
4407:
4401:
4400:
4383:
4377:
4371:
4365:
4359:
4353:
4347:
4341:
4340:
4316:
4310:
4309:
4299:
4290:
4284:
4278:
4272:
4261:
4255:
4249:
4243:
4237:
4231:
4225:
4219:
4213:
4207:
4201:
4166:Möbius monoids.
4153:Dedekind domains
4118:polynomial rings
4109:
4107:
4106:
4101:
4093:
4085:
4080:
4068:
4066:
4065:
4060:
4046:
4034:
4032:
4031:
4026:
4015:
4003:
4001:
4000:
3995:
3993:
3977:
3975:
3974:
3969:
3964:
3952:
3950:
3949:
3944:
3942:
3938:
3930:
3924:
3912:
3910:
3909:
3904:
3902:
3898:
3890:
3884:
3868:
3866:
3865:
3860:
3858:
3854:
3846:
3840:
3820:
3819:
3813:
3812:
3794:
3792:
3791:
3786:
3784:
3780:
3772:
3766:
3751:
3749:
3748:
3743:
3738:
3734:
3733:
3725:
3712:
3708:
3707:
3699:
3658:
3656:
3655:
3650:
3645:
3637:
3632:
3617:
3615:
3614:
3609:
3607:
3606:
3561:
3559:
3558:
3553:
3545:
3544:
3528:
3526:
3525:
3520:
3515:
3510:
3509:
3501:
3489:
3474:
3472:
3471:
3466:
3455:
3424:
3422:
3421:
3416:
3402:
3353:
3346:
3342:
3339:
3333:
3310:
3302:
3293:
3291:
3290:
3285:
3269:
3267:
3266:
3261:
3259:
3258:
3242:
3240:
3239:
3234:
3232:
3231:
3215:
3213:
3212:
3207:
3202:
3201:
3185:
3183:
3182:
3177:
3172:
3171:
3159:
3158:
3143:is a divisor of
3142:
3140:
3139:
3134:
3132:
3131:
3115:
3113:
3112:
3107:
3102:
3101:
3085:
3083:
3082:
3077:
3075:
3074:
3058:
3054:
3050:
3046:
3044:
3043:
3038:
3036:
3035:
3023:
3022:
3006:
3004:
3003:
2998:
2996:
2995:
2979:
2972:
2970:
2969:
2964:
2938:
2937:
2919:
2918:
2906:
2905:
2887:
2886:
2861:
2859:
2858:
2853:
2832:
2830:
2829:
2824:
2819:
2818:
2806:
2805:
2789:
2787:
2786:
2781:
2773:
2772:
2757:
2756:
2734:
2732:
2731:
2726:
2721:
2720:
2708:
2707:
2685:
2683:
2682:
2677:
2675:
2674:
2662:
2661:
2636:
2634:
2633:
2628:
2623:
2622:
2610:
2609:
2593:
2589:
2587:
2586:
2581:
2579:
2578:
2569:
2558:
2557:
2548:
2527:
2525:
2524:
2519:
2514:
2513:
2501:
2500:
2484:
2482:
2481:
2476:
2471:
2470:
2454:
2452:
2451:
2446:
2444:
2443:
2424:
2422:
2421:
2416:
2414:
2407:
2406:
2394:
2393:
2384:
2383:
2368:
2364:
2363:
2351:
2350:
2341:
2340:
2307:
2305:
2304:
2299:
2286:composite number
2283:
2281:
2280:
2275:
2263:
2261:
2260:
2255:
2228:
2222:
2216:
2182:
2176:
2160:
2151:
2142:
2133:
2120:
2109:
2100:
2076:
2067:
2056:
2045:
1995:
1981:
1928:
1900:
1872:
1857:
1847:
1841:
1835:
1829:
1823:
1818:strong induction
1815:
1811:
1750:
1748:
1747:
1742:
1740:
1732:
1728:
1727:
1715:
1714:
1698:
1680:
1675:
1674:
1670:
1669:
1657:
1656:
1636:
1635:
1631:
1630:
1618:
1617:
1597:
1596:
1592:
1591:
1579:
1578:
1558:
1557:
1553:
1552:
1540:
1539:
1483:
1479:
1478:
1466:
1465:
1449:
1431:
1426:
1425:
1421:
1420:
1408:
1407:
1387:
1386:
1382:
1381:
1369:
1368:
1348:
1347:
1343:
1342:
1330:
1329:
1309:
1308:
1304:
1303:
1291:
1290:
1237:
1236:
1235:
1223:
1222:
1212:
1194:
1189:
1188:
1187:
1186:
1174:
1173:
1159:
1158:
1157:
1156:
1144:
1143:
1129:
1128:
1127:
1126:
1114:
1113:
1099:
1098:
1097:
1096:
1084:
1083:
1006:rational numbers
1000:
986:
984:
983:
978:
972:
971:
970:
960:
950:
945:
924:
923:
922:
921:
907:
906:
905:
904:
890:
889:
888:
887:
873:
872:
871:
870:
839:infinite product
836:
832:
826:
806:. For example,
797:
784:
773:
762:
736:
734:
733:
728:
722:
721:
720:
710:
700:
695:
676:
675:
674:
664:
651:
650:
649:
639:
629:
628:
627:
617:
592:
532:
531:, Proposition 14
529:Elements Book IX
514:
513:, Proposition 32
498:infinite descent
492:
491:, Proposition 31
450:
449:, Proposition 30
396:polynomial rings
382:that are called
375:
373:
372:
367:
312:
310:
309:
304:
254:
252:
251:
246:
142:
141:
129:
128:
116:
115:
21:
5238:
5237:
5233:
5232:
5231:
5229:
5228:
5227:
5203:
5202:
5201:
5196:
5140:
5096:
5057:
5028:Highly abundant
5006:
4987:Unitary perfect
4940:
4864:
4855:
4836:Unitary divisor
4814:
4811:
4710:
4701:
4681:
4605:
4589:. Foreword by
4587:J. H. Silverman
4564:
4482:
4464:
4432:
4427:
4419:
4415:
4408:
4404:
4398:
4384:
4380:
4372:
4368:
4360:
4356:
4348:
4344:
4333:
4317:
4313:
4297:
4291:
4287:
4279:
4275:
4262:
4258:
4250:
4246:
4238:
4234:
4226:
4222:
4214:
4210:
4202:
4195:
4191:
4182:Prime signature
4172:
4084:
4076:
4074:
4071:
4070:
4042:
4040:
4037:
4036:
4011:
4009:
4006:
4005:
3989:
3987:
3984:
3983:
3980:integral domain
3960:
3958:
3955:
3954:
3929:
3925:
3920:
3918:
3915:
3914:
3889:
3885:
3880:
3878:
3875:
3874:
3845:
3841:
3836:
3834:
3831:
3830:
3817:
3815:
3810:
3808:
3771:
3767:
3762:
3760:
3757:
3756:
3724:
3717:
3713:
3698:
3691:
3687:
3667:
3664:
3663:
3636:
3628:
3626:
3623:
3622:
3602:
3598:
3575:
3572:
3571:
3540:
3536:
3534:
3531:
3530:
3500:
3490:
3488:
3480:
3477:
3476:
3451:
3449:
3446:
3445:
3398:
3396:
3393:
3392:
3374:complex numbers
3354:
3343:
3337:
3334:
3323:
3311:
3300:
3298:Generalizations
3279:
3276:
3275:
3254:
3250:
3248:
3245:
3244:
3227:
3223:
3221:
3218:
3217:
3197:
3193:
3191:
3188:
3187:
3167:
3163:
3154:
3150:
3148:
3145:
3144:
3127:
3123:
3121:
3118:
3117:
3097:
3093:
3091:
3088:
3087:
3070:
3066:
3064:
3061:
3060:
3056:
3052:
3048:
3031:
3027:
3018:
3014:
3012:
3009:
3008:
2991:
2987:
2985:
2982:
2981:
2977:
2933:
2929:
2914:
2910:
2901:
2897:
2882:
2878:
2870:
2867:
2866:
2838:
2835:
2834:
2814:
2810:
2801:
2797:
2795:
2792:
2791:
2768:
2764:
2752:
2748:
2740:
2737:
2736:
2716:
2712:
2703:
2699:
2691:
2688:
2687:
2670:
2666:
2657:
2653:
2645:
2642:
2641:
2618:
2614:
2605:
2601:
2599:
2596:
2595:
2591:
2574:
2570:
2565:
2553:
2549:
2544:
2533:
2530:
2529:
2509:
2505:
2496:
2492:
2490:
2487:
2486:
2466:
2462:
2460:
2457:
2456:
2439:
2435:
2433:
2430:
2429:
2412:
2411:
2402:
2398:
2389:
2385:
2379:
2375:
2366:
2365:
2359:
2355:
2346:
2342:
2336:
2332:
2325:
2318:
2316:
2313:
2312:
2293:
2290:
2289:
2269:
2266:
2265:
2249:
2246:
2245:
2235:
2224:
2218:
2215:
2206:
2199:
2190:
2184:
2178:
2175:
2168:
2162:
2159:
2153:
2150:
2144:
2141:
2135:
2132:
2126:
2119:
2111:
2108:
2102:
2099:
2090:
2084:
2078:
2075:
2069:
2066:
2058:
2055:
2047:
2044:
2035:
2029:
2022:
2013:
2007:
1997:
1991:
1988:
1980:
1971:
1965:
1959:
1950:
1944:
1930:
1927:
1918:
1912:
1902:
1899:
1890:
1884:
1874:
1859:
1849:
1843:
1837:
1831:
1825:
1821:
1813:
1809:
1806:
1788:The proof uses
1786:
1770:
1764:
1738:
1737:
1723:
1719:
1710:
1706:
1699:
1694:
1679:
1665:
1661:
1652:
1648:
1641:
1637:
1626:
1622:
1613:
1609:
1602:
1598:
1587:
1583:
1574:
1570:
1563:
1559:
1548:
1544:
1535:
1531:
1524:
1520:
1513:
1489:
1488:
1474:
1470:
1461:
1457:
1450:
1445:
1430:
1416:
1412:
1403:
1399:
1392:
1388:
1377:
1373:
1364:
1360:
1353:
1349:
1338:
1334:
1325:
1321:
1314:
1310:
1299:
1295:
1286:
1282:
1275:
1271:
1264:
1243:
1242:
1231:
1227:
1218:
1214:
1213:
1208:
1193:
1182:
1178:
1169:
1165:
1164:
1160:
1152:
1148:
1139:
1135:
1134:
1130:
1122:
1118:
1109:
1105:
1104:
1100:
1092:
1088:
1079:
1075:
1074:
1070:
1063:
1050:
1048:
1045:
1044:
1014:
999:
991:
966:
962:
961:
956:
946:
935:
917:
913:
912:
908:
900:
896:
895:
891:
883:
879:
878:
874:
866:
862:
861:
857:
849:
846:
845:
834:
828:
821:
816:1001 = 7×11×13.
793:
779:
772:
764:
761:
754:
747:
741:
716:
712:
711:
706:
696:
685:
670:
666:
665:
660:
645:
641:
640:
635:
623:
619:
618:
613:
601:
598:
597:
587:
584:
579:
533:
526:
515:
508:
493:
486:
466:divides either
451:
444:
424:
325:
322:
321:
274:
271:
270:
137:
133:
124:
120:
111:
107:
99:
96:
95:
39:
28:
23:
22:
15:
12:
11:
5:
5236:
5226:
5225:
5220:
5215:
5198:
5197:
5195:
5194:
5189:
5184:
5179:
5174:
5169:
5164:
5159:
5154:
5148:
5146:
5142:
5141:
5139:
5138:
5133:
5128:
5123:
5118:
5113:
5107:
5105:
5098:
5097:
5095:
5094:
5089:
5084:
5074:
5068:
5066:
5059:
5058:
5056:
5055:
5050:
5045:
5040:
5035:
5030:
5025:
5020:
5014:
5012:
5008:
5007:
5005:
5004:
4999:
4994:
4989:
4984:
4979:
4974:
4969:
4964:
4959:
4957:Almost perfect
4954:
4948:
4946:
4942:
4941:
4939:
4938:
4933:
4928:
4923:
4918:
4913:
4908:
4903:
4898:
4893:
4888:
4883:
4878:
4872:
4870:
4866:
4865:
4858:
4856:
4854:
4853:
4848:
4843:
4838:
4833:
4828:
4822:
4820:
4816:
4815:
4810:
4809:
4802:
4795:
4787:
4781:
4780:
4762:Grime, James,
4759:
4749:
4731:
4726:
4716:
4709:
4708:External links
4706:
4705:
4704:
4699:
4684:
4679:
4666:
4645:
4624:
4603:
4567:
4562:
4530:Disquisitiones
4522:
4521:
4512:
4486:
4485:
4480:
4467:
4462:
4431:
4428:
4426:
4425:
4413:
4402:
4396:
4378:
4366:
4354:
4342:
4331:
4311:
4285:
4273:
4256:
4244:
4232:
4220:
4208:
4192:
4190:
4187:
4186:
4185:
4179:
4171:
4168:
4099:
4096:
4091:
4088:
4083:
4079:
4058:
4055:
4052:
4049:
4045:
4024:
4021:
4018:
4014:
3992:
3967:
3963:
3941:
3936:
3933:
3928:
3923:
3901:
3896:
3893:
3888:
3883:
3857:
3852:
3849:
3844:
3839:
3799:, then one of
3783:
3778:
3775:
3770:
3765:
3753:
3752:
3741:
3737:
3731:
3728:
3723:
3720:
3716:
3711:
3705:
3702:
3697:
3694:
3690:
3686:
3683:
3680:
3677:
3674:
3671:
3648:
3643:
3640:
3635:
3631:
3605:
3601:
3597:
3594:
3591:
3588:
3585:
3582:
3579:
3551:
3548:
3543:
3539:
3518:
3513:
3507:
3504:
3499:
3496:
3493:
3487:
3484:
3464:
3461:
3458:
3454:
3414:
3411:
3408:
3405:
3401:
3356:
3355:
3314:
3312:
3305:
3299:
3296:
3283:
3257:
3253:
3230:
3226:
3205:
3200:
3196:
3175:
3170:
3166:
3162:
3157:
3153:
3130:
3126:
3105:
3100:
3096:
3073:
3069:
3034:
3030:
3026:
3021:
3017:
2994:
2990:
2974:
2973:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2941:
2936:
2932:
2928:
2925:
2922:
2917:
2913:
2909:
2904:
2900:
2896:
2893:
2890:
2885:
2881:
2877:
2874:
2851:
2848:
2845:
2842:
2822:
2817:
2813:
2809:
2804:
2800:
2779:
2776:
2771:
2767:
2763:
2760:
2755:
2751:
2747:
2744:
2724:
2719:
2715:
2711:
2706:
2702:
2698:
2695:
2673:
2669:
2665:
2660:
2656:
2652:
2649:
2626:
2621:
2617:
2613:
2608:
2604:
2577:
2573:
2568:
2564:
2561:
2556:
2552:
2547:
2543:
2540:
2537:
2517:
2512:
2508:
2504:
2499:
2495:
2474:
2469:
2465:
2442:
2438:
2426:
2425:
2410:
2405:
2401:
2397:
2392:
2388:
2382:
2378:
2374:
2371:
2369:
2367:
2362:
2358:
2354:
2349:
2345:
2339:
2335:
2331:
2328:
2326:
2324:
2321:
2320:
2297:
2273:
2253:
2234:
2231:
2211:
2204:
2195:
2188:
2173:
2166:
2157:
2148:
2139:
2130:
2123:Euclid's lemma
2115:
2106:
2095:
2088:
2082:
2073:
2062:
2051:
2040:
2033:
2027:
2018:
2011:
2005:
1987:
1984:
1976:
1969:
1963:
1955:
1948:
1942:
1923:
1916:
1910:
1895:
1888:
1882:
1824:and less than
1805:
1802:
1790:Euclid's lemma
1785:
1782:
1778:multiplicative
1766:Main article:
1763:
1760:
1752:
1751:
1736:
1731:
1726:
1722:
1718:
1713:
1709:
1705:
1702:
1697:
1693:
1689:
1686:
1683:
1681:
1678:
1673:
1668:
1664:
1660:
1655:
1651:
1647:
1644:
1640:
1634:
1629:
1625:
1621:
1616:
1612:
1608:
1605:
1601:
1595:
1590:
1586:
1582:
1577:
1573:
1569:
1566:
1562:
1556:
1551:
1547:
1543:
1538:
1534:
1530:
1527:
1523:
1519:
1516:
1514:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1490:
1487:
1482:
1477:
1473:
1469:
1464:
1460:
1456:
1453:
1448:
1444:
1440:
1437:
1434:
1432:
1429:
1424:
1419:
1415:
1411:
1406:
1402:
1398:
1395:
1391:
1385:
1380:
1376:
1372:
1367:
1363:
1359:
1356:
1352:
1346:
1341:
1337:
1333:
1328:
1324:
1320:
1317:
1313:
1307:
1302:
1298:
1294:
1289:
1285:
1281:
1278:
1274:
1270:
1267:
1265:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1244:
1241:
1234:
1230:
1226:
1221:
1217:
1211:
1207:
1203:
1200:
1197:
1195:
1192:
1185:
1181:
1177:
1172:
1168:
1163:
1155:
1151:
1147:
1142:
1138:
1133:
1125:
1121:
1117:
1112:
1108:
1103:
1095:
1091:
1087:
1082:
1078:
1073:
1069:
1066:
1064:
1062:
1059:
1056:
1053:
1052:
1013:
1010:
995:
988:
987:
976:
969:
965:
959:
955:
949:
944:
941:
938:
934:
930:
927:
920:
916:
911:
903:
899:
894:
886:
882:
877:
869:
865:
860:
856:
853:
833:(for example,
818:
817:
814:
811:
768:
759:
755:< ... <
752:
745:
738:
737:
726:
719:
715:
709:
705:
699:
694:
691:
688:
684:
680:
673:
669:
663:
659:
655:
648:
644:
638:
634:
626:
622:
616:
612:
608:
605:
583:
580:
578:
575:
559:Article 16 of
524:
506:
484:
476:Euclid's lemma
442:
423:
420:
414:statement and
365:
362:
359:
356:
353:
350:
347:
344:
341:
338:
335:
332:
329:
302:
299:
296:
293:
290:
287:
284:
281:
278:
261:
256:
255:
244:
241:
238:
235:
232:
229:
226:
223:
220:
217:
214:
211:
208:
205:
202:
199:
196:
193:
190:
187:
184:
181:
178:
175:
172:
169:
166:
163:
160:
157:
154:
151:
148:
145:
140:
136:
132:
127:
123:
119:
114:
110:
106:
103:
26:
9:
6:
4:
3:
2:
5235:
5224:
5223:Factorization
5221:
5219:
5216:
5214:
5211:
5210:
5208:
5193:
5190:
5188:
5185:
5183:
5180:
5178:
5175:
5173:
5170:
5168:
5165:
5163:
5160:
5158:
5155:
5153:
5150:
5149:
5147:
5143:
5137:
5134:
5132:
5131:Polydivisible
5129:
5127:
5124:
5122:
5119:
5117:
5114:
5112:
5109:
5108:
5106:
5103:
5099:
5093:
5090:
5088:
5085:
5082:
5078:
5075:
5073:
5070:
5069:
5067:
5064:
5060:
5054:
5051:
5049:
5046:
5044:
5041:
5039:
5036:
5034:
5033:Superabundant
5031:
5029:
5026:
5024:
5021:
5019:
5016:
5015:
5013:
5009:
5003:
5002:Erdős–Nicolas
5000:
4998:
4995:
4993:
4990:
4988:
4985:
4983:
4980:
4978:
4975:
4973:
4970:
4968:
4965:
4963:
4960:
4958:
4955:
4953:
4950:
4949:
4947:
4943:
4937:
4934:
4932:
4929:
4927:
4924:
4922:
4919:
4917:
4914:
4912:
4911:Perfect power
4909:
4907:
4904:
4902:
4899:
4897:
4894:
4892:
4889:
4887:
4884:
4882:
4879:
4877:
4874:
4873:
4871:
4867:
4862:
4852:
4849:
4847:
4844:
4842:
4839:
4837:
4834:
4832:
4829:
4827:
4824:
4823:
4821:
4817:
4808:
4803:
4801:
4796:
4794:
4789:
4788:
4785:
4777:
4773:
4769:
4765:
4760:
4757:
4753:
4750:
4747:
4743:
4739:
4735:
4732:
4730:
4727:
4724:
4720:
4717:
4715:
4712:
4711:
4702:
4696:
4692:
4691:
4685:
4682:
4680:0-8176-3743-5
4676:
4672:
4667:
4663:
4659:
4655:
4654:Prentice Hall
4651:
4646:
4642:
4638:
4634:
4630:
4625:
4622:
4618:
4614:
4610:
4606:
4600:
4596:
4592:
4588:
4584:
4581:, Revised by
4580:
4576:
4575:Wright, E. M.
4572:
4568:
4565:
4559:
4555:
4551:
4546:
4545:
4539:
4535:
4534:
4533:
4531:
4527:
4518:
4513:
4509:
4504:
4503:
4502:
4500:
4496:
4492:
4483:
4481:0-8284-0191-8
4477:
4473:
4468:
4465:
4459:
4455:
4451:
4450:
4444:
4443:
4442:
4439:
4438:
4422:
4417:
4411:
4406:
4399:
4397:9783319173689
4393:
4389:
4382:
4375:
4370:
4364:, p. 55)
4363:
4358:
4352:, p. 45)
4351:
4346:
4339:
4334:
4332:9781134977246
4328:
4324:
4323:
4315:
4308:
4303:
4296:
4289:
4282:
4277:
4270:
4266:
4260:
4253:
4248:
4242:, p. 53)
4241:
4236:
4230:, p. 44)
4229:
4224:
4217:
4212:
4205:
4200:
4198:
4193:
4183:
4180:
4177:
4174:
4173:
4167:
4163:
4161:
4156:
4154:
4150:
4146:
4142:
4138:
4133:
4131:
4127:
4123:
4119:
4115:
4110:
4097:
4089:
4086:
4056:
4050:
4019:
3981:
3965:
3939:
3934:
3931:
3926:
3899:
3894:
3891:
3886:
3872:
3855:
3850:
3847:
3842:
3828:
3824:
3806:
3802:
3798:
3781:
3776:
3773:
3768:
3739:
3735:
3729:
3726:
3721:
3718:
3714:
3709:
3703:
3700:
3695:
3692:
3688:
3684:
3681:
3678:
3675:
3672:
3669:
3662:
3661:
3660:
3641:
3638:
3619:
3603:
3599:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3569:
3565:
3564:root of unity
3549:
3546:
3541:
3537:
3516:
3511:
3505:
3502:
3497:
3494:
3491:
3485:
3482:
3459:
3443:
3439:
3434:
3432:
3428:
3412:
3406:
3390:
3386:
3382:
3378:
3375:
3371:
3367:
3363:
3352:
3349:
3341:
3331:
3327:
3321:
3320:
3315:This section
3313:
3309:
3304:
3303:
3295:
3281:
3271:
3255:
3251:
3228:
3224:
3203:
3198:
3194:
3173:
3168:
3164:
3160:
3155:
3151:
3128:
3124:
3103:
3098:
3094:
3071:
3067:
3032:
3028:
3024:
3019:
3015:
2992:
2988:
2960:
2957:
2954:
2948:
2945:
2942:
2934:
2930:
2926:
2923:
2915:
2911:
2907:
2902:
2898:
2891:
2888:
2883:
2879:
2875:
2872:
2865:
2864:
2863:
2849:
2846:
2843:
2840:
2820:
2815:
2811:
2807:
2802:
2798:
2777:
2774:
2769:
2765:
2761:
2758:
2753:
2749:
2745:
2742:
2722:
2717:
2713:
2709:
2704:
2700:
2696:
2693:
2671:
2667:
2663:
2658:
2654:
2650:
2647:
2638:
2624:
2619:
2615:
2611:
2606:
2602:
2575:
2571:
2566:
2562:
2559:
2554:
2550:
2545:
2541:
2538:
2535:
2515:
2510:
2506:
2502:
2497:
2493:
2472:
2467:
2463:
2440:
2436:
2408:
2403:
2399:
2395:
2390:
2386:
2380:
2376:
2372:
2370:
2360:
2356:
2352:
2347:
2343:
2337:
2333:
2329:
2327:
2322:
2311:
2310:
2309:
2295:
2288:greater than
2287:
2271:
2251:
2242:
2240:
2230:
2227:
2221:
2214:
2210:
2203:
2198:
2194:
2187:
2181:
2172:
2165:
2156:
2147:
2138:
2129:
2124:
2118:
2114:
2110:divides some
2105:
2098:
2094:
2087:
2081:
2072:
2065:
2061:
2054:
2050:
2046:, where each
2043:
2039:
2032:
2026:
2021:
2017:
2010:
2004:
2000:
1994:
1983:
1979:
1975:
1968:
1962:
1958:
1954:
1947:
1941:
1937:
1933:
1926:
1922:
1915:
1909:
1905:
1898:
1894:
1887:
1881:
1877:
1871:
1867:
1863:
1856:
1852:
1846:
1840:
1834:
1828:
1819:
1801:
1799:
1795:
1791:
1781:
1779:
1775:
1769:
1759:
1757:
1734:
1724:
1720:
1716:
1711:
1707:
1695:
1691:
1687:
1684:
1682:
1676:
1666:
1662:
1658:
1653:
1649:
1638:
1627:
1623:
1619:
1614:
1610:
1599:
1588:
1584:
1580:
1575:
1571:
1560:
1549:
1545:
1541:
1536:
1532:
1521:
1517:
1515:
1507:
1504:
1501:
1495:
1492:
1485:
1475:
1471:
1467:
1462:
1458:
1446:
1442:
1438:
1435:
1433:
1427:
1417:
1413:
1409:
1404:
1400:
1389:
1378:
1374:
1370:
1365:
1361:
1350:
1339:
1335:
1331:
1326:
1322:
1311:
1300:
1296:
1292:
1287:
1283:
1272:
1268:
1266:
1258:
1255:
1252:
1239:
1232:
1228:
1224:
1219:
1215:
1209:
1205:
1201:
1198:
1196:
1190:
1183:
1179:
1175:
1170:
1166:
1161:
1153:
1149:
1145:
1140:
1136:
1131:
1123:
1119:
1115:
1110:
1106:
1101:
1093:
1089:
1085:
1080:
1076:
1071:
1067:
1065:
1060:
1057:
1054:
1043:
1042:
1041:
1039:
1035:
1031:
1027:
1023:
1019:
1009:
1007:
1002:
998:
994:
974:
967:
963:
957:
953:
942:
939:
936:
932:
928:
925:
918:
914:
909:
901:
897:
892:
884:
880:
875:
867:
863:
858:
854:
851:
844:
843:
842:
840:
831:
824:
815:
812:
809:
808:
807:
805:
801:
800:standard form
796:
791:
786:
782:
777:
776:empty product
771:
767:
758:
751:
744:
724:
717:
713:
707:
703:
697:
692:
689:
686:
682:
678:
671:
667:
661:
657:
653:
646:
642:
636:
632:
624:
620:
614:
610:
606:
603:
596:
595:
594:
590:
574:
572:
568:
567:
562:
557:
555:
551:
546:
544:
539:
530:
523:
519:
512:
505:
501:
499:
490:
483:
479:
477:
473:
469:
465:
461:
457:
448:
441:
437:
435:
434:
429:
419:
417:
416:Wiles's proof
413:
409:
405:
401:
397:
393:
389:
385:
381:
376:
363:
360:
357:
354:
351:
348:
345:
342:
339:
336:
333:
330:
327:
319:
314:
300:
297:
294:
291:
288:
285:
282:
279:
276:
268:
263:
259:
242:
239:
236:
233:
230:
227:
224:
221:
218:
215:
212:
209:
206:
203:
200:
197:
191:
188:
185:
179:
176:
173:
167:
164:
161:
158:
155:
152:
149:
143:
138:
134:
130:
125:
121:
117:
112:
108:
104:
101:
94:
93:
92:
90:
86:
85:prime numbers
82:
78:
74:
70:
66:
58:
54:
50:
49:
43:
37:
33:
19:
5192:Superperfect
5187:Refactorable
4982:Superperfect
4977:Hyperperfect
4962:Quasiperfect
4850:
4846:Prime factor
4767:
4746:Andrew Wiles
4723:cut-the-knot
4688:
4670:
4649:
4628:
4591:Andrew Wiles
4578:
4571:Hardy, G. H.
4543:
4529:
4525:
4523:
4516:
4507:
4498:
4494:
4490:
4487:
4471:
4448:
4435:
4433:
4416:
4405:
4387:
4381:
4369:
4357:
4345:
4336:
4321:
4314:
4307:proposition.
4305:
4301:
4288:
4276:
4259:
4247:
4235:
4223:
4211:
4164:
4157:
4141:ideal number
4134:
4111:
3825:2 is called
3804:
3800:
3796:
3754:
3620:
3435:
3426:
3388:
3384:
3380:
3376:
3359:
3344:
3338:January 2024
3335:
3324:Please help
3319:verification
3316:
3272:
2975:
2790:Also, since
2639:
2427:
2308:. Now, say
2244:Assume that
2243:
2236:
2225:
2219:
2212:
2208:
2201:
2196:
2192:
2185:
2179:
2170:
2163:
2154:
2145:
2136:
2127:
2116:
2112:
2103:
2096:
2092:
2085:
2079:
2070:
2063:
2059:
2052:
2048:
2041:
2037:
2030:
2024:
2019:
2015:
2008:
2002:
1998:
1992:
1989:
1977:
1973:
1966:
1960:
1956:
1952:
1945:
1939:
1935:
1931:
1924:
1920:
1913:
1907:
1903:
1896:
1892:
1885:
1879:
1875:
1869:
1865:
1861:
1854:
1850:
1844:
1838:
1832:
1826:
1807:
1793:
1787:
1771:
1753:
1040:themselves:
1037:
1033:
1029:
1025:
1015:
1003:
996:
992:
989:
835:1000 = 2×3×5
829:
822:
819:
803:
799:
794:
789:
787:
780:
769:
765:
756:
749:
742:
739:
588:
585:
577:Applications
564:
558:
547:
535:
521:
517:
503:
495:
481:
471:
467:
463:
459:
455:
453:
439:
431:
425:
386:and include
377:
315:
264:
257:
76:
72:
68:
62:
46:
5116:Extravagant
5111:Equidigital
5072:Untouchable
4992:Semiperfect
4972:Hemiperfect
4901:Square-free
4772:Brady Haran
4768:Numberphile
4218:, Art. 131)
4216:Gauss (1986
4204:Gauss (1986
4069:but not in
3827:irreducible
1020:(GCD), and
813:1000 = 2×5,
810:999 = 3×37,
65:mathematics
5207:Categories
5152:Arithmetic
5145:Other sets
5104:-dependent
4621:1159.11001
4430:References
4350:Long (1972
4281:Weil (2007
4228:Long (1972
4206:, Art. 16)
3562:is a cube
3442:Eisenstein
1986:Uniqueness
543:André Weil
5182:Descartes
5157:Deficient
5092:Betrothed
4997:Practical
4886:Semiprime
4881:Composite
4641:77-171950
4577:(2008) ,
4423:, § 14.6)
4087:−
4051:ω
3932:−
3892:−
3848:−
3774:−
3727:−
3722:−
3701:−
3679:⋅
3639:−
3600:ω
3596:±
3590:ω
3587:±
3578:±
3538:ω
3503:−
3492:−
3483:ω
3460:ω
3161:−
3025:−
2946:−
2908:−
2876:−
2710:⋯
2664:⋯
2396:⋯
2353:⋯
1848:, where
1804:Existence
1754:However,
1688:∏
1677:⋯
1496:
1439:∏
1428:⋯
1202:∏
1191:⋯
1058:⋅
948:∞
933:∏
926:⋯
798:, or the
683:∏
654:⋯
364:…
355:⋅
349:⋅
337:⋅
298:⋅
286:⋅
243:…
234:⋅
228:⋅
222:⋅
216:⋅
210:⋅
204:⋅
189:⋅
180:⋅
174:⋅
165:⋅
159:⋅
153:⋅
131:⋅
118:⋅
5167:Solitary
5162:Friendly
5087:Sociable
5077:Amicable
5065:-related
5018:Abundant
4916:Achilles
4906:Powerful
4819:Overview
4776:archived
4662:77-81766
4540:(1956),
4454:Springer
4376:, § 1.2)
4254:, Thm 2)
4170:See also
4145:Dedekind
4135:In 1843
3818:−5
3811:−5
3475:, where
2833:one has
2735:one has
2640:Setting
2143:. Since
2134:divides
2077:divides
1794:Elements
1774:additive
820:Factors
527:Euclid,
525:—
509:Euclid,
507:—
487:Euclid,
485:—
445:Euclid,
443:—
433:Elements
412:Fermat's
5172:Sublime
5126:Harshad
4952:Perfect
4936:Unusual
4926:Regular
4896:Sphenic
4831:Divisor
4758:, 2007.
4613:2445243
4304:: 209.
3816:√
3809:√
3529:
1860:1 <
1798:divides
462:, then
422:History
398:over a
81:integer
51:(1801)
5121:Frugal
5081:Triple
4921:Smooth
4891:Pronic
4697:
4677:
4660:
4639:
4619:
4611:
4601:
4560:
4538:Euclid
4478:
4460:
4394:
4329:
4269:ideals
4149:ideals
4137:Kummer
3383:where
2428:Every
1858:, and
740:where
591:> 1
550:Euclid
548:While
428:Euclid
394:, and
67:, the
5136:Smith
5053:Weird
4931:Rough
4876:Prime
4740:from
4554:Dover
4526:Werke
4298:(PDF)
4263:In a
4189:Notes
4122:field
3871:prime
3431:up to
2101:, so
1868:<
1830:. If
1784:Proof
748:<
561:Gauss
400:field
89:up to
53:Gauss
5102:Base
4695:ISBN
4675:ISBN
4658:LCCN
4637:LCCN
4599:ISBN
4585:and
4558:ISBN
4476:ISBN
4458:ISBN
4434:The
4392:ISBN
4327:ISBN
4128:and
4035:and
3387:and
3366:ring
3243:and
2955:<
2844:<
2808:<
2686:and
2612:<
2207:...
2191:...
2152:and
2091:...
2057:and
2036:...
2014:...
1972:⋅⋅⋅
1951:⋅⋅⋅
1919:⋅⋅⋅
1901:and
1891:⋅⋅⋅
1842:and
1776:and
1036:and
1028:and
102:1200
75:and
4721:at
4617:Zbl
4501:".
3873:in
3829:in
3803:or
3368:of
3328:by
3047:or
2241:.
2121:by
1936:a b
1855:a b
1701:max
1643:max
1604:max
1565:max
1526:max
1493:lcm
1452:min
1394:min
1355:min
1316:min
1277:min
1247:gcd
825:= 1
802:of
792:of
785:).
783:= 0
563:'s
470:or
430:'s
313:).
260:can
63:In
45:In
34:or
5209::
4774:,
4770:,
4766:,
4656:,
4635:,
4615:,
4609:MR
4607:,
4597:,
4573:;
4556:,
4532:.
4456:,
4335:.
4300:.
4196:^
4155:.
4132:.
4124:,
3797:ab
3440:,
3381:bi
3379:+
2229:.
2200:=
2169:=
2023:=
2001:=
1938:=
1934:=
1906:=
1878:=
1864:≤
1853:=
1008:.
573:.
500:.
460:ab
436:.
418:.
390:,
277:12
87:,
5083:)
5079:(
4806:e
4799:t
4792:v
4748:.
4725:.
4665:.
4644:.
4499:n
4491:n
4271:.
4098:.
4095:]
4090:5
4082:[
4078:Z
4057:,
4054:]
4048:[
4044:Z
4023:]
4020:i
4017:[
4013:Z
3991:Z
3966:.
3962:Z
3940:]
3935:5
3927:[
3922:Z
3900:]
3895:5
3887:[
3882:Z
3856:]
3851:5
3843:[
3838:Z
3805:b
3801:a
3782:]
3777:5
3769:[
3764:Z
3740:.
3736:)
3730:5
3719:1
3715:(
3710:)
3704:5
3696:+
3693:1
3689:(
3685:=
3682:3
3676:2
3673:=
3670:6
3647:]
3642:5
3634:[
3630:Z
3604:2
3593:,
3584:,
3581:1
3550:1
3547:=
3542:3
3517:,
3512:2
3506:3
3498:+
3495:1
3486:=
3463:]
3457:[
3453:Z
3427:i
3413:.
3410:]
3407:i
3404:[
3400:Z
3389:b
3385:a
3377:a
3351:)
3345:(
3340:)
3336:(
3322:.
3282:1
3256:1
3252:q
3229:1
3225:p
3204:,
3199:1
3195:q
3174:,
3169:1
3165:p
3156:1
3152:q
3129:1
3125:p
3104:.
3099:j
3095:q
3072:1
3068:p
3057:s
3053:Q
3049:Q
3033:1
3029:p
3020:1
3016:q
2993:1
2989:p
2978:s
2961:.
2958:s
2952:)
2949:Q
2943:P
2940:(
2935:1
2931:p
2927:=
2924:Q
2921:)
2916:1
2912:p
2903:1
2899:q
2895:(
2892:=
2889:Q
2884:1
2880:p
2873:s
2850:.
2847:P
2841:Q
2821:,
2816:1
2812:q
2803:1
2799:p
2778:.
2775:Q
2770:1
2766:q
2762:=
2759:P
2754:1
2750:p
2746:=
2743:s
2723:,
2718:n
2714:q
2705:2
2701:q
2697:=
2694:Q
2672:m
2668:p
2659:2
2655:p
2651:=
2648:P
2625:,
2620:1
2616:q
2607:1
2603:p
2592:s
2576:j
2572:q
2567:/
2563:s
2560:=
2555:i
2551:p
2546:/
2542:s
2539:=
2536:t
2516:,
2511:j
2507:q
2503:=
2498:i
2494:p
2473:.
2468:j
2464:q
2441:i
2437:p
2409:.
2404:n
2400:q
2391:2
2387:q
2381:1
2377:q
2373:=
2361:m
2357:p
2348:2
2344:p
2338:1
2334:p
2330:=
2323:s
2296:1
2272:s
2252:s
2226:n
2220:n
2213:k
2209:q
2205:2
2202:q
2197:j
2193:p
2189:2
2186:p
2180:n
2174:1
2171:q
2167:1
2164:p
2158:1
2155:q
2149:1
2146:p
2140:1
2137:q
2131:1
2128:p
2117:i
2113:q
2107:1
2104:p
2097:k
2093:q
2089:2
2086:q
2083:1
2080:q
2074:1
2071:p
2064:i
2060:q
2053:i
2049:p
2042:k
2038:q
2034:2
2031:q
2028:1
2025:q
2020:j
2016:p
2012:2
2009:p
2006:1
2003:p
1999:n
1993:n
1978:k
1974:q
1970:2
1967:q
1964:1
1961:q
1957:j
1953:p
1949:2
1946:p
1943:1
1940:p
1932:n
1925:k
1921:q
1917:2
1914:q
1911:1
1908:q
1904:b
1897:j
1893:p
1889:2
1886:p
1883:1
1880:p
1876:a
1870:n
1866:b
1862:a
1851:n
1845:b
1839:a
1833:n
1827:n
1822:1
1814:2
1810:1
1792:(
1735:.
1730:)
1725:i
1721:b
1717:,
1712:i
1708:a
1704:(
1696:i
1692:p
1685:=
1672:)
1667:4
1663:b
1659:,
1654:4
1650:a
1646:(
1639:7
1633:)
1628:3
1624:b
1620:,
1615:3
1611:a
1607:(
1600:5
1594:)
1589:2
1585:b
1581:,
1576:2
1572:a
1568:(
1561:3
1555:)
1550:1
1546:b
1542:,
1537:1
1533:a
1529:(
1522:2
1518:=
1511:)
1508:b
1505:,
1502:a
1499:(
1486:,
1481:)
1476:i
1472:b
1468:,
1463:i
1459:a
1455:(
1447:i
1443:p
1436:=
1423:)
1418:4
1414:b
1410:,
1405:4
1401:a
1397:(
1390:7
1384:)
1379:3
1375:b
1371:,
1366:3
1362:a
1358:(
1351:5
1345:)
1340:2
1336:b
1332:,
1327:2
1323:a
1319:(
1312:3
1306:)
1301:1
1297:b
1293:,
1288:1
1284:a
1280:(
1273:2
1269:=
1262:)
1259:b
1256:,
1253:a
1250:(
1240:,
1233:i
1229:b
1225:+
1220:i
1216:a
1210:i
1206:p
1199:=
1184:4
1180:b
1176:+
1171:4
1167:a
1162:7
1154:3
1150:b
1146:+
1141:3
1137:a
1132:5
1124:2
1120:b
1116:+
1111:2
1107:a
1102:3
1094:1
1090:b
1086:+
1081:1
1077:a
1072:2
1068:=
1061:b
1055:a
1038:b
1034:a
1030:b
1026:a
997:i
993:n
975:,
968:i
964:n
958:i
954:p
943:1
940:=
937:i
929:=
919:4
915:n
910:7
902:3
898:n
893:5
885:2
881:n
876:3
868:1
864:n
859:2
855:=
852:n
830:n
823:p
804:n
795:n
781:k
770:i
766:n
760:k
757:p
753:2
750:p
746:1
743:p
725:,
718:i
714:n
708:i
704:p
698:k
693:1
690:=
687:i
679:=
672:k
668:n
662:k
658:p
647:2
643:n
637:2
633:p
625:1
621:n
615:1
611:p
607:=
604:n
589:n
472:b
468:a
464:p
456:p
361:=
358:1
352:1
346:2
343:=
340:1
334:2
331:=
328:2
301:4
295:3
292:=
289:6
283:2
280:=
240:=
237:2
231:2
225:3
219:2
213:5
207:2
201:5
198:=
195:)
192:5
186:5
183:(
177:3
171:)
168:2
162:2
156:2
150:2
147:(
144:=
139:2
135:5
126:1
122:3
113:4
109:2
105:=
59:.
38:.
20:)
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