36:
8071:. Lagrange was the first to realize that "a coherent general theory required the simulatenous consideration of all forms." He was the first to recognize the importance of the discriminant and to define the essential notions of equivalence and reduction, which, according to Weil, have "dominated the whole subject of quadratic forms ever since". Lagrange showed that there are finitely many equivalence classes of given discriminant, thereby defining for the first time an arithmetic
6392:"Composition" also sometimes refers to, roughly, a binary operation on binary quadratic forms. The word "roughly" indicates two caveats: only certain pairs of binary quadratic forms can be composed, and the resulting form is not well-defined (although its equivalence class is). The composition operation on equivalence classes is defined by first defining composition of forms and then showing that this induces a well-defined operation on classes.
7054:
8100:. Gauss introduced a very general version of a composition operator that allows composing even forms of different discriminants and imprimitive forms. He replaced Lagrange's equivalence with the more precise notion of proper equivalence, and this enabled him to show that the primitive classes of given discriminant form a
4248:
5509:
7996:
of forms. Each genus is the union of a finite number of equivalence classes of the same discriminant, with the number of classes depending only on the discriminant. In the context of binary quadratic forms, genera can be defined either through congruence classes of numbers represented by forms or by
8017:
There is circumstantial evidence of protohistoric knowledge of algebraic identities involving binary quadratic forms. The first problem concerning binary quadratic forms asks for the existence or construction of representations of integers by particular binary quadratic forms. The prime examples
6826:
8171:
Even so, work on binary quadratic forms with integer coefficients continues to the present. This includes numerous results about quadratic number fields, which can often be translated into the language of binary quadratic forms, but also includes developments about forms themselves or that
6410:
A variety of definitions of composition of forms has been given, often in an attempt to simplify the extremely technical and general definition of Gauss. We present here Arndt's method, because it remains rather general while being simple enough to be amenable to computations by hand. An
1077:
6395:"Composition" can also refer to a binary operation on representations of integers by forms. This operation is substantially more complicated than composition of forms, but arose first historically. We will consider such operations in a separate section below.
4062:
5372:
8124:
These investigations of Gauss strongly influenced both the arithmetical theory of quadratic forms in more than two variables and the subsequent development of algebraic number theory, where quadratic fields are replaced with more general
7273:
to be the smallest positive solution to the system of congruences above. Alternatively, we may view the result of composition, not as a form, but as an equivalence class of forms modulo the action of the group of matrices of the form
2766:
5281:
2046:
1574:
5557:
on the set of representations of integers by binary quadratic forms. It follows that equivalence defined this way is an equivalence relation and in particular that the forms in equivalent representations are equivalent forms.
7049:{\displaystyle {\begin{aligned}x&\equiv B_{1}{\pmod {2{\tfrac {A_{1}}{e}}}}\\x&\equiv B_{2}{\pmod {2{\tfrac {A_{2}}{e}}}}\\{\tfrac {B_{\mu }}{e}}x&\equiv {\tfrac {\Delta +B_{1}B_{2}}{2e}}{\pmod {2A}}\end{aligned}}}
687:
Binary quadratic forms are closely related to ideals in quadratic fields. This allows the class number of a quadratic field to be calculated by counting the number of reduced binary quadratic forms of a given discriminant.
1657:
6212:, which ever since has been the reduction algorithm most commonly given in textbooks. In 1981, Zagier published an alternative reduction algorithm which has found several uses as an alternative to Gauss's.
5725:
2242:
773:
7328:
8104:
under the composition operation. He introduced genus theory, which gives a powerful way to understand the quotient of the class group by the subgroup of squares. (Gauss and many subsequent authors wrote
3917:
958:
5364:
2180:
950:
887:
4336:. "Describe" can mean various things: give an algorithm to generate all representations, a closed formula for the number of representations, or even just determine whether any representations exist.
7746:
1382:
627:
6831:
6600:
6508:
4067:
2675:
963:
6688:
1503:
5880:
6323:
5555:
2341:
2092:
1704:
6817:
4765:
2839:
1979:
6756:
188:
8133:
contains truly revolutionary ideas and involves very complicated computations, sometimes left to the reader. Combined, the novelty and complexity made
Section V notoriously difficult.
3973:
3659:
333:
4941:
1771:
1141:
7982:
7924:
7866:
7808:
7394:
7201:
7135:
1823:
1440:
7696:
5787:
5663:
5225:
5170:
4879:
The minimum absolute value represented by a class is zero for degenerate classes and positive for definite and indefinite classes. All numbers represented by a definite form
4243:{\displaystyle {\begin{aligned}(3\cdot 17+4\cdot 12,2\cdot 17+3\cdot 12)&=(99,70),\\(3\cdot 99+4\cdot 70,2\cdot 99+3\cdot 70)&=(577,408),\\&\vdots \end{aligned}}}
5929:
5608:
4054:
3740:
3549:
2294:
682:
529:
480:
431:
382:
8139:
5504:{\displaystyle {\begin{pmatrix}\delta &-\beta \\-\gamma &\alpha \end{pmatrix}}{\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}}={\begin{pmatrix}x_{2}\\y_{2}\end{pmatrix}}}
3796:
3341:
3272:
2994:
2636:
291:
8067:
6044:
5994:
5103:
4503:
4420:
4294:
1892:
8452:
5060:
4666:
4583:
4543:
4460:
4377:
3607:
3485:
3381:
3125:
3030:
8096:
5996:, every representation is equivalent to a unique representation by a reduced form, so a complete set of representatives is given by the finitely many representations of
1251:
1170:
2503:
1222:
6109:
4852:
4804:
4622:
1196:
7637:
6620:
6387:
6363:
6343:
6277:
6253:
6165:
6068:
6018:
5960:
1916:
1862:
6139:
4993:
4967:
3065:
2396:
808:
7617:
7590:
7563:
7536:
7509:
7482:
7455:
7428:
7255:
7228:
4005:
3828:
3691:
3157:
3226:
2662:
3461:
3431:
2945:
7079:
4330:
3401:
3295:
3197:
3177:
3085:
2919:
2899:
2879:
2859:
2590:
2570:
2550:
2459:
2439:
2419:
2670:
232:
This article is entirely devoted to integral binary quadratic forms. This choice is motivated by their status as the driving force behind the development of
8087:, which summarized the work of Euler and Lagrange and added some of his own contributions, including the first glimpse of a composition operation on forms.
6398:
Composition means taking 2 quadratic forms of the same discriminant and combining them to create a quadratic form of the same discriminant, as follows from
5233:
1998:
1526:
1505:. Since Gauss it has been recognized that this definition is inferior to that given above. If there is a need to distinguish, sometimes forms are called
8156:
4545:
at all. In the first case, the sixteen representations were explicitly described. It was also shown that the number of representations of an integer by
8075:. His introduction of reduction allowed the quick enumeration of the classes of given discriminant and foreshadowed the eventual development of
6259:
have since become one of the central ideas in algebraic number theory. From a modern perspective, the class group of a fundamental discriminant
6227:
on primitive equivalence classes of forms of the same discriminant, one of the deepest discoveries of Gauss, which makes this set into a finite
8058:
provided the first proofs of Fermat's observations and added some new conjectures about representations by specific forms, without proof.
4872:
The minimum absolute value represented by a class. This is the smallest nonnegative value in the set of integers represented by a class.
1582:
8206:
8051:
5671:
2188:
1072:{\displaystyle {\begin{aligned}f(\alpha x+\beta y,\gamma x+\delta y)&=g(x,y),\\\alpha \delta -\beta \gamma &=1.\end{aligned}}}
694:
7280:
8005:
in n variables. This states that forms are in the same genus if they are locally equivalent at all rational primes (including the
3836:
5289:
2105:
912:
8137:
published simplifications of the theory that made it accessible to a broader audience. The culmination of this work is his text
6167:. The set of all such representations constitutes a complete set of representatives for equivalence classes of representations.
813:
8026:
in the 7th century CE. Several centuries later, his ideas were extended to a complete solution of Pell's equation known as the
8022:
and the representation of integers as sums of two squares. Pell's equation was already considered by the Indian mathematician
629:
and so on are quadratic forms, and the theory of quadratic forms gives a unified way of looking at and proving these theorems.
7701:
1260:
8557:
8512:
8475:
1706:
on the set of binary quadratic forms. The equivalence relation above then arises from the general theory of group actions.
534:
236:. Since the late nineteenth century, binary quadratic forms have given up their preeminence in algebraic number theory to
6513:
6421:
6629:
1467:
1461:
can mean either a function defined on equivalence classes of forms or a property shared by all forms in the same class.
5792:
8400:
6293:
5517:
2303:
2054:
1666:
79:
57:
6761:
4678:
3559:
in a solution gives another solution, so it is enough to seek just solutions in positive integers. One solution is
2774:
1844:
Terminology has arisen for classifying classes and their forms in terms of their invariants. A form of discriminant
50:
7698:, i.e., with first coefficient 1. (It can be shown that all such forms lie in a single class, and the restriction
1933:
6693:
8050:
made several observations about representations by specific quadratic forms including that which is now known as
7396:
under this action, the middle coefficients of the forms in the class form a congruence class of integers modulo 2
110:
8076:
8629:
8459:
4253:
These values will keep growing in size, so we see there are infinitely many ways to represent 1 by the form
3926:
8619:
5963:
3612:
296:
8624:
7400:. Thus, composition gives a well-defined function from pairs of binary quadratic forms to such classes.
6208:
5882:. This is the recursion step in the process described above for generating infinitely many solutions to
4882:
1712:
5665:. Such a representation is a solution to the Pell equation described in the examples above. The matrix
1085:
8504:
7929:
7871:
7813:
7755:
7341:
7148:
7095:
1783:
1660:
1387:
8006:
7646:
7619:. It follows that composition induces a well-defined operation on primitive classes of discriminant
7265:. One way to make this a well-defined operation is to make an arbitrary convention for how to choose
5737:
5613:
5175:
5120:
8216:
8002:
6399:
5885:
5564:
4010:
3696:
3505:
2250:
638:
485:
436:
387:
338:
8645:
8038:. The problem of representing integers by sums of two squares was considered in the 3rd century by
8031:
3745:
3300:
3231:
2953:
2595:
1927:
250:
44:
6023:
5973:
5065:
4465:
4382:
4256:
1871:
8417:
8165:
8161:
5025:
4631:
4586:
4548:
4508:
4425:
4342:
3562:
3466:
3346:
3090:
3002:
233:
7257:. We see that its first coefficient is well-defined, but the other two depend on the choice of
1227:
1146:
2464:
1201:
101:
61:
6073:
4821:
4773:
4591:
1926:
if its content is 1, that is, if its coefficients are coprime. If a form's discriminant is a
1464:
Lagrange used a different notion of equivalence, in which the second condition is replaced by
1175:
8080:
8023:
7622:
6605:
6372:
6348:
6328:
6262:
6238:
6144:
6053:
6003:
5945:
1901:
1847:
6118:
4972:
4946:
3035:
2366:
778:
8567:
8522:
8485:
8091:
7595:
7568:
7541:
7514:
7487:
7460:
7433:
7406:
7233:
7206:
4339:
The examples above discuss the representation problem for the numbers 3 and 65 by the form
4297:
3978:
3801:
3664:
3130:
1517:
1446:
8612:
8575:
8492:
5931:. Iterating this matrix action, we find that the infinite set of representations of 1 by
4869:
is represented by a form in a class, then it is represented by all other forms in a class.
3202:
2641:
8:
8545:
8118:
8101:
3436:
3406:
2924:
7061:
2761:{\displaystyle {\begin{aligned}65&=1^{2}+8^{2},\\65&=4^{2}+7^{2},\end{aligned}}}
8181:
8019:
7992:
Gauss also considered a coarser notion of equivalence, with each coarse class called a
7639:, and as mentioned above, Gauss showed these classes form a finite abelian group. The
6284:
5938:
There are generally finitely many equivalence classes of representations of an integer
4815:
4315:
3491:
3386:
3280:
3182:
3162:
3070:
2904:
2884:
2864:
2844:
2575:
2555:
2535:
2444:
2424:
2404:
1450:
633:
213:, most results are not specific to the case of two variables, so they are described in
8553:
8541:
8508:
8471:
8396:
8072:
8027:
6366:
5276:{\displaystyle {\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}}
2041:{\displaystyle {\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}}
1569:{\displaystyle {\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}}
8599:
Zetafunktionen und quadratische Körper: eine Einführung in die höhere
Zahlentheorie
8571:
8496:
7640:
6224:
6179:, there are only finitely many classes of binary quadratic forms with discriminant
4875:
The congruence classes modulo the discriminant of a class represented by the class.
8563:
8518:
8481:
8467:
8211:
8185:
7749:
6288:
3228:
or 1. We can check these nine pairs directly to see that none of them satisfies
1449:
on the set of integral quadratic forms. It follows that the quadratic forms are
237:
2921:. In all, there are sixteen different solution pairs. On the other hand, when
4296:. This recursive description was discussed in Theon of Smyrna's commentary on
214:
210:
24:
8035:
2841:
that do the trick. We obtain more pairs that work by switching the values of
8639:
8201:
8189:
8173:
7058:
It can be shown that this system always has a unique integer solution modulo
6412:
6228:
8582:
8549:
8533:
8126:
4861:
There are several class invariants relevant to the representation problem:
2352:
1778:
244:, but advances specific to binary quadratic forms still occur on occasion.
241:
8001:
defined on the set of forms. A third definition is a special case of the
8529:
8148:
8043:
6256:
93:
8594:
8177:
8160:, the theory of binary quadratic forms lost its preeminent position in
8039:
6280:
3487:. There are only a finite number of pairs satisfying this constraint.
2529:
8613:
Peter
Luschny, Positive numbers represented by a binary quadratic form
23:
coefficients. For binary quadratic forms with other coefficients, see
8587:
Number Theory: An approach through history from
Hammurapi to Legendre
8134:
8144:
8062:
3490:
Another ancient problem involving quadratic forms asks us to solve
2297:
1652:{\displaystyle f(x,y)\mapsto f(\alpha x+\beta y,\gamma x+\delta y)}
8466:, Graduate Texts in Mathematics, vol. 138, Berlin, New York:
7748:
implies that there exists such a form of every discriminant.) To
4308:
The oldest problem in the theory of binary quadratic forms is the
8152:
6199:, for constructing a canonical representative in each class, the
5022:
is indefinite. We saw instances of this in the examples above:
4807:
247:
Pierre Fermat stated that if p is an odd prime then the equation
218:
20:
6046:, Zagier proved that every representation of a positive integer
5720:{\displaystyle {\begin{pmatrix}3&-4\\-2&3\end{pmatrix}}}
2237:{\displaystyle {\begin{pmatrix}3&-4\\-2&3\end{pmatrix}}}
768:{\displaystyle \sum _{(m,n)\in \mathbb {Z} ^{2}}q^{m^{2}+n^{2}}}
8047:
8143:. The third edition of this work includes two supplements by
7323:{\displaystyle {\begin{pmatrix}1&n\\0&1\end{pmatrix}}}
8151:, and from then on, especially after the 1897 publication of
8055:
3912:{\displaystyle (3\cdot 3+4\cdot 2,2\cdot 3+3\cdot 2)=(17,12)}
8042:. In the 17th century, inspired while reading Diophantus's
7643:
class in the group is the unique class containing all forms
5359:{\displaystyle f(\alpha x+\beta y,\gamma x+\delta y)=g(x,y)}
2175:{\displaystyle f(\alpha x+\beta y,\gamma x+\delta y)=f(x,y)}
945:{\displaystyle \alpha ,\beta ,\gamma ,{\text{ and }}\delta }
8503:, Cambridge Studies in Advanced Mathematics, vol. 27,
6203:, whose coefficients are the smallest in a suitable sense.
4865:
The set of integers represented by a class. If an integer
2999:
does not have integer solutions. To see why, we note that
882:{\displaystyle \sum _{(m,n)\in \mathbb {Z} ^{2}}q^{f(m,n)}}
8454:, Fermat, class field theory, and complex multiplication
7741:{\displaystyle \Delta \equiv 0{\text{ or }}1{\pmod {4}}}
5514:
The above conditions give a (right) action of the group
1377:{\displaystyle g=(-3x+2y)^{2}+4(-3x+2y)(x-y)+2(x-y)^{2}}
8061:
The general theory of quadratic forms was initiated by
8164:
and became overshadowed by the more general theory of
7289:
6982:
6953:
6926:
6867:
6772:
6647:
5680:
5466:
5423:
5381:
5242:
5113:
The notion of equivalence of forms can be extended to
2197:
2007:
1535:
8420:
7932:
7874:
7816:
7758:
7704:
7649:
7625:
7598:
7571:
7544:
7517:
7490:
7463:
7436:
7409:
7344:
7283:
7236:
7209:
7151:
7098:
7081:. We arbitrarily choose such a solution and call it
7064:
6829:
6764:
6696:
6632:
6608:
6516:
6424:
6375:
6351:
6331:
6296:
6265:
6241:
6147:
6121:
6076:
6056:
6026:
6006:
5976:
5948:
5888:
5795:
5740:
5674:
5616:
5567:
5520:
5375:
5292:
5236:
5178:
5123:
5068:
5028:
4975:
4949:
4885:
4824:
4776:
4681:
4634:
4594:
4551:
4511:
4505:
in infinitely many ways and 3 is not represented by
4468:
4462:
in sixteen different ways, while 1 is represented by
4428:
4385:
4345:
4318:
4259:
4065:
4013:
3981:
3929:
3839:
3804:
3748:
3699:
3667:
3615:
3565:
3508:
3469:
3439:
3409:
3389:
3349:
3303:
3283:
3234:
3205:
3185:
3165:
3133:
3093:
3073:
3038:
3005:
2956:
2927:
2907:
2887:
2867:
2847:
2777:
2673:
2644:
2598:
2578:
2558:
2538:
2467:
2447:
2427:
2407:
2369:
2306:
2253:
2191:
2108:
2057:
2001:
1936:
1904:
1874:
1850:
1828:
The content, equal to the greatest common divisor of
1786:
1715:
1669:
1585:
1529:
1470:
1390:
1263:
1230:
1204:
1178:
1149:
1088:
961:
915:
816:
781:
697:
641:
622:{\displaystyle x^{2}+y^{2},x^{2}+2y^{2},x^{2}-3y^{2}}
537:
488:
439:
390:
341:
299:
253:
113:
8113:; the modern convention allowing the coefficient of
8030:, attributed to either of the Indian mathematicians
5789:
by this matrix yields the equivalent representation
1930:, then the form is primitive. Discriminants satisfy
1520:
terminology, which is used occasionally below, when
335:, and he made similar statement about the equations
8129:. But the impact was not immediate. Section V of
7987:
6595:{\displaystyle f_{2}=A_{2}x^{2}+B_{2}xy+C_{2}y^{2}}
6503:{\displaystyle f_{1}=A_{1}x^{2}+B_{1}xy+C_{1}y^{2}}
3798:is another such pair. For instance, from the pair
3274:, so the equation does not have integer solutions.
8446:
7976:
7918:
7860:
7802:
7740:
7690:
7631:
7611:
7584:
7557:
7530:
7503:
7476:
7449:
7422:
7388:
7322:
7249:
7222:
7195:
7129:
7073:
7048:
6811:
6750:
6683:{\displaystyle B_{\mu }={\tfrac {B_{1}+B_{2}}{2}}}
6682:
6614:
6594:
6502:
6381:
6357:
6337:
6317:
6271:
6247:
6159:
6133:
6103:
6062:
6038:
6012:
5988:
5954:
5923:
5874:
5781:
5719:
5657:
5602:
5549:
5503:
5358:
5275:
5219:
5164:
5097:
5054:
4987:
4961:
4935:
4846:
4798:
4759:
4660:
4616:
4577:
4537:
4497:
4454:
4414:
4371:
4324:
4288:
4242:
4048:
3999:
3967:
3911:
3822:
3790:
3734:
3685:
3653:
3601:
3543:
3479:
3455:
3425:
3395:
3375:
3335:
3289:
3266:
3220:
3191:
3171:
3151:
3119:
3079:
3059:
3024:
2988:
2939:
2913:
2893:
2873:
2853:
2833:
2760:
2656:
2630:
2584:
2564:
2544:
2497:
2453:
2433:
2413:
2390:
2335:
2288:
2236:
2174:
2086:
2040:
1973:
1910:
1886:
1856:
1817:
1765:
1698:
1651:
1568:
1498:{\displaystyle \alpha \delta -\beta \gamma =\pm 1}
1497:
1434:
1376:
1245:
1216:
1190:
1164:
1135:
1071:
944:
881:
802:
767:
676:
621:
523:
474:
425:
376:
327:
285:
182:
19:This article is about binary quadratic forms with
8464:A Course in Computational Algebraic Number Theory
4312:: describe the representations of a given number
3975:. Iterating this process, we find further pairs
3343:can have only a finite number of solutions since
16:Quadratic homogeneous polynomial in two variables
8637:
6703:
5875:{\displaystyle 1=f(3x_{1}+4y_{1},2x_{1}+3y_{1})}
6318:{\displaystyle \mathbf {Q} ({\sqrt {\Delta }})}
5935:that were determined above are all equivalent.
5550:{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )}
5286:with integer entries and determinant 1 so that
2336:{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )}
2087:{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )}
1699:{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )}
1579:has integer entries and determinant 1, the map
691:The classical theta function of 2 variables is
8617:
8491:
8292:
8172:originated by thinking about forms, including
8090:The theory was vastly extended and refined by
6812:{\displaystyle A={\tfrac {A_{1}A_{2}}{e^{2}}}}
6602:, each primitive and of the same discriminant
6405:
6170:
4760:{\displaystyle r_{2}(n)=4(d_{1}(n)-d_{3}(n)),}
2881:and/or by changing the sign of one or both of
2834:{\displaystyle (x,y)=(1,8){\text{ and }}(4,7)}
6206:Gauss gave a superior reduction algorithm in
5108:
4303:
1974:{\displaystyle \Delta \equiv 0,1{\pmod {4}}.}
7338:is an integer. If we consider the class of
6751:{\displaystyle e=\gcd(A_{1},A_{2},B_{\mu })}
6365:, the narrow class group is the same as the
5730:has determinant 1 and is an automorphism of
5006:The number of representations of an integer
1513:if they are equivalent in Lagrange's sense.
8552:. (6th ed.), Oxford: Clarendon Press,
8528:
8258:
7868:. Alternatively, we can form the class of
5966:for these classes can be given in terms of
2347:is definite, the group is finite, and when
1445:The above equivalence conditions define an
810:is a positive definite quadratic form then
183:{\displaystyle q(x,y)=ax^{2}+bxy+cy^{2},\,}
4995:. For this reason, the former are called
6070:is equivalent to a unique representation
5540:
2326:
2077:
1689:
952:such that the following conditions hold:
842:
723:
209:. When the coefficients can be arbitrary
179:
80:Learn how and when to remove this message
8538:An Introduction to the Theory of Numbers
8188:reinterpretation of composition through
6235:(or simply class group) of discriminant
5227:are equivalent if there exists a matrix
43:This article includes a list of general
8207:Fermat's theorem on sums of two squares
8052:Fermat's theorem on sums of two squares
6411:alternative definition is described at
5942:by forms of given nonzero discriminant
4624:gives the number of representations of
3277:A similar argument shows that for each
2532:considered whether, for an odd integer
632:Another instance of quadratic forms is
8638:
8593:
8281:
8270:
7511:respectively, then the composition of
3968:{\displaystyle 1=17^{2}-2\cdot 12^{2}}
3494:. For instance, we may seek integers
8458:
8234:
6175:Lagrange proved that for every value
3923:and we can check that this satisfies
2296:. The automorphisms of a form are a
8581:
8376:
8364:
8352:
8340:
8328:
8316:
8304:
8246:
7565:is equivalent to the composition of
6195:. He described an algorithm, called
5970:defined in the section below. When
4422:. We see that 65 is represented by
3654:{\displaystyle 1=3^{2}-2\cdot 2^{2}}
1773:, then important invariants include
328:{\displaystyle p\equiv 1{\pmod {4}}}
29:
8391:Johannes Buchmann, Ulrich Vollmer:
7730:
7031:
6921:
6862:
6622:. We perform the following steps:
4936:{\displaystyle f=ax^{2}+bxy+cy^{2}}
2421:if it is possible to find integers
1960:
1766:{\displaystyle f=ax^{2}+bxy+cy^{2}}
317:
13:
7752:a class, we take a representative
7705:
7626:
7099:
6985:
6609:
6376:
6352:
6332:
6307:
6266:
6242:
6057:
6027:
6007:
5977:
5949:
5526:
5523:
4858:that are congruent to 3 modulo 4.
2552:, it is possible to find integers
2351:is indefinite, it is infinite and
2312:
2309:
2063:
2060:
1937:
1905:
1875:
1851:
1787:
1675:
1672:
1136:{\displaystyle f=x^{2}+4xy+2y^{2}}
49:it lacks sufficient corresponding
14:
8657:
8606:
7977:{\displaystyle Ax^{2}-Bxy+Cy^{2}}
7919:{\displaystyle Cx^{2}+Bxy+Ay^{2}}
7861:{\displaystyle Ax^{2}-Bxy+Cy^{2}}
7803:{\displaystyle Ax^{2}+Bxy+Cy^{2}}
7389:{\displaystyle Ax^{2}+Bxy+Cy^{2}}
7196:{\displaystyle Ax^{2}+Bxy+Cy^{2}}
7130:{\displaystyle \Delta =B^{2}-4AC}
6418:Suppose we wish to compose forms
6115:is reduced in Zagier's sense and
6000:by reduced forms of discriminant
4379:and for the number 1 by the form
2358:
1818:{\displaystyle \Delta =b^{2}-4ac}
1453:into equivalence classes, called
1435:{\displaystyle -x^{2}+4xy-2y^{2}}
8085:Essai sur la théorie des nombres
7988:Genera of binary quadratic forms
7691:{\displaystyle x^{2}+Bxy+Cy^{2}}
6821:Solve the system of congruences
6298:
5782:{\displaystyle 1=f(x_{1},y_{1})}
5734:. Acting on the representation
5658:{\displaystyle 1=f(x_{1},y_{1})}
5220:{\displaystyle n=g(x_{2},y_{2})}
5165:{\displaystyle m=f(x_{1},y_{1})}
4943:have the same sign: positive if
3609:, that is, there is an equality
1983:
1509:using the definition above and
34:
8370:
8358:
8346:
8334:
7723:
7024:
6914:
6855:
2247:is an automorphism of the form
1953:
310:
8322:
8310:
8298:
8286:
8275:
8264:
8252:
8240:
8228:
8140:Vorlesungen über Zahlentheorie
7734:
7724:
7038:
7025:
6944:
6915:
6885:
6856:
6745:
6706:
6312:
6302:
6215:
6098:
6086:
5924:{\displaystyle 1=x^{2}-2y^{2}}
5869:
5805:
5776:
5750:
5652:
5626:
5610:and consider a representation
5603:{\displaystyle f=x^{2}-2y^{2}}
5544:
5536:
5353:
5341:
5332:
5296:
5214:
5188:
5159:
5133:
4841:
4835:
4793:
4787:
4751:
4748:
4742:
4726:
4720:
4707:
4698:
4692:
4611:
4605:
4220:
4208:
4198:
4150:
4140:
4128:
4118:
4070:
4049:{\displaystyle 1=x^{2}-2y^{2}}
3994:
3982:
3906:
3894:
3888:
3840:
3817:
3805:
3785:
3749:
3735:{\displaystyle 1=x^{2}-2y^{2}}
3680:
3668:
3596:
3584:
3578:
3566:
3544:{\displaystyle 1=x^{2}-2y^{2}}
3449:
3441:
3419:
3411:
3159:is one of the nine pairs with
3146:
3134:
2828:
2816:
2808:
2796:
2790:
2778:
2489:
2477:
2385:
2373:
2330:
2322:
2289:{\displaystyle f=x^{2}-2y^{2}}
2169:
2157:
2148:
2112:
2081:
2073:
1992:is a quadratic form, a matrix
1964:
1954:
1693:
1685:
1646:
1610:
1604:
1601:
1589:
1365:
1352:
1343:
1331:
1328:
1307:
1292:
1270:
1030:
1018:
1005:
969:
892:
874:
862:
834:
822:
797:
785:
715:
703:
677:{\displaystyle x^{2}-ny^{2}=1}
524:{\displaystyle p=x^{2}-3y^{2}}
475:{\displaystyle p=x^{2}-2y^{2}}
426:{\displaystyle p=x^{2}+3y^{2}}
377:{\displaystyle p=x^{2}+2y^{2}}
321:
311:
223:integral binary quadratic form
129:
117:
1:
8385:
4854:is the number of divisors of
4672:. There is a closed formula
3791:{\displaystyle (3x+4y,2x+3y)}
3336:{\displaystyle n=x^{2}+y^{2}}
3267:{\displaystyle 3=x^{2}+y^{2}}
2989:{\displaystyle 3=x^{2}+y^{2}}
2631:{\displaystyle n=x^{2}+y^{2}}
286:{\displaystyle p=x^{2}+y^{2}}
8147:. Supplement XI introduces
6039:{\displaystyle \Delta >0}
5989:{\displaystyle \Delta <0}
5098:{\displaystyle x^{2}-2y^{2}}
5018:is definite and infinite if
4498:{\displaystyle x^{2}-2y^{2}}
4415:{\displaystyle x^{2}-2y^{2}}
4289:{\displaystyle x^{2}-2y^{2}}
1887:{\displaystyle \Delta <0}
7:
8625:Encyclopedia of Mathematics
8447:{\displaystyle x^{2}+y^{2}}
8195:
8097:Disquisitiones Arithmeticae
6406:Composing forms and classes
6209:Disquisitiones Arithmeticae
6171:Reduction and class numbers
5055:{\displaystyle x^{2}+y^{2}}
4661:{\displaystyle x^{2}+y^{2}}
4578:{\displaystyle x^{2}+y^{2}}
4538:{\displaystyle x^{2}+y^{2}}
4455:{\displaystyle x^{2}+y^{2}}
4372:{\displaystyle x^{2}+y^{2}}
3602:{\displaystyle (x,y)=(3,2)}
3480:{\displaystyle {\sqrt {n}}}
3403:unless the absolute values
3376:{\displaystyle x^{2}+y^{2}}
3120:{\displaystyle x^{2}+y^{2}}
3025:{\displaystyle x^{2}\geq 4}
2524:
2182:. For example, the matrix
10:
8662:
8505:Cambridge University Press
8293:Fröhlich & Taylor 1993
8012:
6223:most commonly refers to a
6050:by a form of discriminant
5115:equivalent representations
5109:Equivalent representations
4332:by a given quadratic form
4304:The representation problem
1246:{\displaystyle \delta =-1}
1165:{\displaystyle \alpha =-3}
221:coefficients is called an
18:
8409:, Springer, New York 1989
8395:, Springer, Berlin 2007,
8068:Recherches d'Arithmétique
8003:genus of a quadratic form
5062:is positive definite and
4999:forms and the latter are
2498:{\displaystyle n=q(x,y).}
1918:is a perfect square, and
1217:{\displaystyle \gamma =1}
217:. A quadratic form with
8618:A. V. Malyshev (2001) ,
8222:
7403:It can be shown that if
7203:is "the" composition of
7137:. It can be shown that
6389:it may be twice as big.
6104:{\displaystyle n=f(x,y)}
4847:{\displaystyle d_{3}(n)}
4799:{\displaystyle d_{1}(n)}
4617:{\displaystyle r_{2}(n)}
2461:satisfying the equation
2363:A binary quadratic form
1928:fundamental discriminant
1191:{\displaystyle \beta =2}
909:if there exist integers
8620:"Binary quadratic form"
8501:Algebraic number theory
8259:Hardy & Wright 2008
8166:algebraic number fields
8162:algebraic number theory
7632:{\displaystyle \Delta }
6615:{\displaystyle \Delta }
6382:{\displaystyle \Delta }
6358:{\displaystyle \Delta }
6338:{\displaystyle \Delta }
6272:{\displaystyle \Delta }
6248:{\displaystyle \Delta }
6160:{\displaystyle y\geq 0}
6063:{\displaystyle \Delta }
6013:{\displaystyle \Delta }
5955:{\displaystyle \Delta }
4587:sum of squares function
4585:is always finite. The
1911:{\displaystyle \Delta }
1857:{\displaystyle \Delta }
234:algebraic number theory
225:, often abbreviated to
64:more precise citations.
8448:
8407:Binary Quadratic Forms
8393:Binary Quadratic Forms
8217:Brahmagupta's identity
7978:
7920:
7862:
7810:and form the class of
7804:
7742:
7692:
7633:
7613:
7586:
7559:
7532:
7505:
7478:
7451:
7424:
7390:
7324:
7269:—for instance, choose
7251:
7224:
7197:
7131:
7075:
7057:
7050:
6813:
6752:
6684:
6616:
6596:
6504:
6400:Brahmagupta's identity
6383:
6359:
6339:
6319:
6273:
6249:
6183:. Their number is the
6161:
6135:
6134:{\displaystyle x>0}
6105:
6064:
6040:
6014:
5990:
5956:
5925:
5876:
5783:
5721:
5659:
5604:
5551:
5505:
5360:
5277:
5221:
5166:
5099:
5056:
4989:
4988:{\displaystyle a<0}
4963:
4962:{\displaystyle a>0}
4937:
4848:
4800:
4761:
4662:
4618:
4579:
4539:
4499:
4456:
4416:
4373:
4326:
4310:representation problem
4290:
4244:
4050:
4001:
3969:
3913:
3824:
3792:
3736:
3687:
3655:
3603:
3545:
3481:
3457:
3427:
3397:
3377:
3337:
3291:
3268:
3222:
3193:
3173:
3153:
3121:
3081:
3061:
3060:{\displaystyle x=-1,0}
3026:
2990:
2941:
2915:
2895:
2875:
2855:
2835:
2762:
2658:
2632:
2586:
2566:
2546:
2505:Such an equation is a
2499:
2455:
2435:
2415:
2392:
2391:{\displaystyle q(x,y)}
2337:
2290:
2238:
2176:
2088:
2042:
1975:
1912:
1888:
1858:
1819:
1767:
1700:
1653:
1570:
1499:
1457:of quadratic forms. A
1436:
1384:, which simplifies to
1378:
1247:
1218:
1192:
1166:
1137:
1073:
946:
883:
804:
803:{\displaystyle f(x,y)}
769:
678:
623:
525:
476:
427:
378:
329:
287:
184:
102:homogeneous polynomial
8449:
8180:reduction algorithm,
7979:
7921:
7863:
7805:
7743:
7693:
7634:
7614:
7612:{\displaystyle g_{2}}
7587:
7585:{\displaystyle g_{1}}
7560:
7558:{\displaystyle f_{2}}
7533:
7531:{\displaystyle f_{1}}
7506:
7504:{\displaystyle g_{2}}
7479:
7477:{\displaystyle g_{1}}
7452:
7450:{\displaystyle f_{2}}
7425:
7423:{\displaystyle f_{1}}
7391:
7325:
7252:
7250:{\displaystyle f_{2}}
7225:
7223:{\displaystyle f_{1}}
7198:
7132:
7076:
7051:
6822:
6814:
6753:
6685:
6617:
6597:
6505:
6384:
6360:
6340:
6320:
6274:
6250:
6162:
6136:
6106:
6065:
6041:
6015:
5991:
5962:. A complete set of
5957:
5926:
5877:
5784:
5722:
5660:
5605:
5552:
5506:
5361:
5278:
5222:
5167:
5100:
5057:
4990:
4964:
4938:
4849:
4801:
4762:
4663:
4619:
4580:
4540:
4500:
4457:
4417:
4374:
4327:
4291:
4245:
4051:
4002:
4000:{\displaystyle (x,y)}
3970:
3914:
3825:
3823:{\displaystyle (3,2)}
3793:
3737:
3688:
3686:{\displaystyle (x,y)}
3656:
3604:
3551:. Changing signs of
3546:
3482:
3458:
3428:
3398:
3378:
3338:
3292:
3269:
3223:
3194:
3174:
3154:
3152:{\displaystyle (x,y)}
3127:will exceed 3 unless
3122:
3082:
3062:
3027:
2991:
2942:
2916:
2896:
2876:
2856:
2836:
2763:
2659:
2633:
2587:
2567:
2547:
2500:
2456:
2436:
2416:
2393:
2338:
2291:
2239:
2177:
2089:
2043:
1976:
1922:otherwise. A form is
1913:
1889:
1859:
1820:
1768:
1701:
1654:
1571:
1511:improperly equivalent
1500:
1437:
1379:
1248:
1219:
1193:
1167:
1138:
1074:
947:
889:is a theta function.
884:
805:
770:
679:
624:
526:
477:
428:
379:
330:
288:
227:binary quadratic form
185:
98:binary quadratic form
8418:
8117:to be odd is due to
8018:are the solution of
7930:
7872:
7814:
7756:
7702:
7647:
7623:
7596:
7569:
7542:
7515:
7488:
7461:
7434:
7407:
7342:
7281:
7234:
7207:
7149:
7096:
7062:
6827:
6762:
6694:
6630:
6606:
6514:
6422:
6373:
6349:
6329:
6294:
6263:
6239:
6145:
6119:
6074:
6054:
6024:
6004:
5974:
5946:
5886:
5793:
5738:
5672:
5614:
5565:
5518:
5373:
5290:
5234:
5176:
5121:
5066:
5026:
4973:
4947:
4883:
4822:
4774:
4679:
4632:
4592:
4549:
4509:
4466:
4426:
4383:
4343:
4316:
4257:
4063:
4011:
3979:
3927:
3837:
3802:
3746:
3697:
3665:
3613:
3563:
3506:
3467:
3437:
3407:
3387:
3347:
3301:
3281:
3232:
3221:{\displaystyle -1,0}
3203:
3183:
3163:
3131:
3091:
3071:
3036:
3003:
2954:
2925:
2905:
2885:
2865:
2845:
2775:
2671:
2657:{\displaystyle n=65}
2642:
2596:
2576:
2556:
2536:
2465:
2445:
2425:
2405:
2367:
2304:
2251:
2189:
2106:
2055:
1999:
1934:
1902:
1872:
1848:
1784:
1713:
1667:
1583:
1527:
1468:
1447:equivalence relation
1388:
1261:
1228:
1202:
1176:
1147:
1086:
959:
913:
814:
779:
695:
639:
535:
486:
437:
388:
339:
297:
251:
111:
8589:, Birkhäuser Boston
8414:Primes of the form
6369:, but for positive
5561:As an example, let
5117:. Representations
3693:is any solution to
3463:are both less than
3456:{\displaystyle |y|}
3426:{\displaystyle |x|}
2940:{\displaystyle n=3}
1507:properly equivalent
293:has a solution iff
8493:Fröhlich, Albrecht
8444:
7974:
7916:
7858:
7800:
7738:
7688:
7629:
7609:
7582:
7555:
7528:
7501:
7474:
7457:are equivalent to
7447:
7420:
7386:
7320:
7314:
7247:
7220:
7193:
7127:
7074:{\displaystyle 2A}
7071:
7046:
7044:
7021:
6969:
6942:
6883:
6809:
6807:
6748:
6680:
6678:
6612:
6592:
6500:
6379:
6355:
6335:
6315:
6285:narrow class group
6269:
6245:
6157:
6131:
6101:
6060:
6036:
6010:
5986:
5952:
5921:
5872:
5779:
5717:
5711:
5655:
5600:
5547:
5501:
5495:
5452:
5412:
5356:
5273:
5267:
5217:
5162:
5095:
5052:
4985:
4959:
4933:
4844:
4818:to 1 modulo 4 and
4796:
4757:
4658:
4614:
4575:
4535:
4495:
4452:
4412:
4369:
4322:
4286:
4240:
4238:
4046:
3997:
3965:
3909:
3820:
3788:
3732:
3683:
3651:
3599:
3541:
3477:
3453:
3423:
3393:
3373:
3333:
3287:
3264:
3218:
3189:
3169:
3149:
3117:
3077:
3057:
3022:
2986:
2937:
2911:
2891:
2871:
2851:
2831:
2758:
2756:
2654:
2628:
2582:
2562:
2542:
2495:
2451:
2431:
2411:
2388:
2333:
2286:
2234:
2228:
2172:
2084:
2038:
2032:
1971:
1908:
1884:
1854:
1815:
1763:
1696:
1649:
1566:
1560:
1495:
1432:
1374:
1243:
1214:
1188:
1162:
1133:
1082:For example, with
1069:
1067:
942:
879:
853:
800:
765:
734:
674:
619:
521:
472:
423:
374:
325:
283:
180:
8559:978-0-19-921986-5
8542:D. R. Heath-Brown
8514:978-0-521-43834-6
8477:978-3-540-55640-4
8405:Duncan A. Buell:
8355:, Ch.III §§VII-IX
8343:, Ch.II §§VIII-XI
8307:, Ch.I §§VI, VIII
8028:chakravala method
8007:Archimedean place
7717:
7020:
6968:
6941:
6882:
6806:
6677:
6367:ideal class group
6310:
5001:negative definite
4997:positive definite
4806:is the number of
4668:as a function of
4325:{\displaystyle n}
4298:Euclid's Elements
3475:
3396:{\displaystyle n}
3290:{\displaystyle n}
3192:{\displaystyle y}
3172:{\displaystyle x}
3080:{\displaystyle 1}
2914:{\displaystyle y}
2894:{\displaystyle x}
2874:{\displaystyle y}
2854:{\displaystyle x}
2814:
2771:so we find pairs
2585:{\displaystyle y}
2565:{\displaystyle x}
2545:{\displaystyle n}
2454:{\displaystyle y}
2434:{\displaystyle x}
2414:{\displaystyle n}
1257:is equivalent to
937:
817:
698:
240:and more general
104:in two variables
90:
89:
82:
8653:
8632:
8602:
8590:
8578:
8525:
8488:
8453:
8451:
8450:
8445:
8443:
8442:
8430:
8429:
8380:
8374:
8368:
8362:
8356:
8350:
8344:
8338:
8332:
8326:
8320:
8314:
8308:
8302:
8296:
8290:
8284:
8279:
8273:
8268:
8262:
8256:
8250:
8244:
8238:
8232:
8184:topographs, and
8176:infrastructure,
8094:in Section V of
7999:genus characters
7984:are equivalent.
7983:
7981:
7980:
7975:
7973:
7972:
7945:
7944:
7925:
7923:
7922:
7917:
7915:
7914:
7887:
7886:
7867:
7865:
7864:
7859:
7857:
7856:
7829:
7828:
7809:
7807:
7806:
7801:
7799:
7798:
7771:
7770:
7747:
7745:
7744:
7739:
7737:
7718:
7715:
7697:
7695:
7694:
7689:
7687:
7686:
7659:
7658:
7638:
7636:
7635:
7630:
7618:
7616:
7615:
7610:
7608:
7607:
7591:
7589:
7588:
7583:
7581:
7580:
7564:
7562:
7561:
7556:
7554:
7553:
7537:
7535:
7534:
7529:
7527:
7526:
7510:
7508:
7507:
7502:
7500:
7499:
7483:
7481:
7480:
7475:
7473:
7472:
7456:
7454:
7453:
7448:
7446:
7445:
7429:
7427:
7426:
7421:
7419:
7418:
7395:
7393:
7392:
7387:
7385:
7384:
7357:
7356:
7329:
7327:
7326:
7321:
7319:
7318:
7256:
7254:
7253:
7248:
7246:
7245:
7229:
7227:
7226:
7221:
7219:
7218:
7202:
7200:
7199:
7194:
7192:
7191:
7164:
7163:
7136:
7134:
7133:
7128:
7114:
7113:
7080:
7078:
7077:
7072:
7055:
7053:
7052:
7047:
7045:
7041:
7022:
7019:
7011:
7010:
7009:
7000:
6999:
6983:
6970:
6964:
6963:
6954:
6947:
6943:
6937:
6936:
6927:
6912:
6911:
6888:
6884:
6878:
6877:
6868:
6853:
6852:
6818:
6816:
6815:
6810:
6808:
6805:
6804:
6795:
6794:
6793:
6784:
6783:
6773:
6757:
6755:
6754:
6749:
6744:
6743:
6731:
6730:
6718:
6717:
6689:
6687:
6686:
6681:
6679:
6673:
6672:
6671:
6659:
6658:
6648:
6642:
6641:
6621:
6619:
6618:
6613:
6601:
6599:
6598:
6593:
6591:
6590:
6581:
6580:
6562:
6561:
6549:
6548:
6539:
6538:
6526:
6525:
6509:
6507:
6506:
6501:
6499:
6498:
6489:
6488:
6470:
6469:
6457:
6456:
6447:
6446:
6434:
6433:
6388:
6386:
6385:
6380:
6364:
6362:
6361:
6356:
6344:
6342:
6341:
6336:
6325:of discriminant
6324:
6322:
6321:
6316:
6311:
6306:
6301:
6278:
6276:
6275:
6270:
6254:
6252:
6251:
6246:
6233:form class group
6225:binary operation
6191:of discriminant
6189:
6188:
6166:
6164:
6163:
6158:
6140:
6138:
6137:
6132:
6110:
6108:
6107:
6102:
6069:
6067:
6066:
6061:
6045:
6043:
6042:
6037:
6019:
6017:
6016:
6011:
5995:
5993:
5992:
5987:
5961:
5959:
5958:
5953:
5930:
5928:
5927:
5922:
5920:
5919:
5904:
5903:
5881:
5879:
5878:
5873:
5868:
5867:
5852:
5851:
5836:
5835:
5820:
5819:
5788:
5786:
5785:
5780:
5775:
5774:
5762:
5761:
5726:
5724:
5723:
5718:
5716:
5715:
5664:
5662:
5661:
5656:
5651:
5650:
5638:
5637:
5609:
5607:
5606:
5601:
5599:
5598:
5583:
5582:
5556:
5554:
5553:
5548:
5543:
5535:
5534:
5529:
5510:
5508:
5507:
5502:
5500:
5499:
5492:
5491:
5478:
5477:
5457:
5456:
5449:
5448:
5435:
5434:
5417:
5416:
5365:
5363:
5362:
5357:
5282:
5280:
5279:
5274:
5272:
5271:
5226:
5224:
5223:
5218:
5213:
5212:
5200:
5199:
5171:
5169:
5168:
5163:
5158:
5157:
5145:
5144:
5104:
5102:
5101:
5096:
5094:
5093:
5078:
5077:
5061:
5059:
5058:
5053:
5051:
5050:
5038:
5037:
4994:
4992:
4991:
4986:
4969:and negative if
4968:
4966:
4965:
4960:
4942:
4940:
4939:
4934:
4932:
4931:
4904:
4903:
4853:
4851:
4850:
4845:
4834:
4833:
4805:
4803:
4802:
4797:
4786:
4785:
4766:
4764:
4763:
4758:
4741:
4740:
4719:
4718:
4691:
4690:
4667:
4665:
4664:
4659:
4657:
4656:
4644:
4643:
4623:
4621:
4620:
4615:
4604:
4603:
4584:
4582:
4581:
4576:
4574:
4573:
4561:
4560:
4544:
4542:
4541:
4536:
4534:
4533:
4521:
4520:
4504:
4502:
4501:
4496:
4494:
4493:
4478:
4477:
4461:
4459:
4458:
4453:
4451:
4450:
4438:
4437:
4421:
4419:
4418:
4413:
4411:
4410:
4395:
4394:
4378:
4376:
4375:
4370:
4368:
4367:
4355:
4354:
4331:
4329:
4328:
4323:
4295:
4293:
4292:
4287:
4285:
4284:
4269:
4268:
4249:
4247:
4246:
4241:
4239:
4229:
4055:
4053:
4052:
4047:
4045:
4044:
4029:
4028:
4006:
4004:
4003:
3998:
3974:
3972:
3971:
3966:
3964:
3963:
3945:
3944:
3918:
3916:
3915:
3910:
3829:
3827:
3826:
3821:
3797:
3795:
3794:
3789:
3741:
3739:
3738:
3733:
3731:
3730:
3715:
3714:
3692:
3690:
3689:
3684:
3660:
3658:
3657:
3652:
3650:
3649:
3631:
3630:
3608:
3606:
3605:
3600:
3550:
3548:
3547:
3542:
3540:
3539:
3524:
3523:
3486:
3484:
3483:
3478:
3476:
3471:
3462:
3460:
3459:
3454:
3452:
3444:
3432:
3430:
3429:
3424:
3422:
3414:
3402:
3400:
3399:
3394:
3382:
3380:
3379:
3374:
3372:
3371:
3359:
3358:
3342:
3340:
3339:
3334:
3332:
3331:
3319:
3318:
3296:
3294:
3293:
3288:
3273:
3271:
3270:
3265:
3263:
3262:
3250:
3249:
3227:
3225:
3224:
3219:
3198:
3196:
3195:
3190:
3178:
3176:
3175:
3170:
3158:
3156:
3155:
3150:
3126:
3124:
3123:
3118:
3116:
3115:
3103:
3102:
3086:
3084:
3083:
3078:
3066:
3064:
3063:
3058:
3031:
3029:
3028:
3023:
3015:
3014:
2995:
2993:
2992:
2987:
2985:
2984:
2972:
2971:
2946:
2944:
2943:
2938:
2920:
2918:
2917:
2912:
2900:
2898:
2897:
2892:
2880:
2878:
2877:
2872:
2860:
2858:
2857:
2852:
2840:
2838:
2837:
2832:
2815:
2812:
2767:
2765:
2764:
2759:
2757:
2750:
2749:
2737:
2736:
2710:
2709:
2697:
2696:
2663:
2661:
2660:
2655:
2637:
2635:
2634:
2629:
2627:
2626:
2614:
2613:
2591:
2589:
2588:
2583:
2571:
2569:
2568:
2563:
2551:
2549:
2548:
2543:
2520:
2514:
2504:
2502:
2501:
2496:
2460:
2458:
2457:
2452:
2440:
2438:
2437:
2432:
2420:
2418:
2417:
2412:
2397:
2395:
2394:
2389:
2342:
2340:
2339:
2334:
2329:
2321:
2320:
2315:
2295:
2293:
2292:
2287:
2285:
2284:
2269:
2268:
2243:
2241:
2240:
2235:
2233:
2232:
2181:
2179:
2178:
2173:
2093:
2091:
2090:
2085:
2080:
2072:
2071:
2066:
2047:
2045:
2044:
2039:
2037:
2036:
1980:
1978:
1977:
1972:
1967:
1917:
1915:
1914:
1909:
1893:
1891:
1890:
1885:
1863:
1861:
1860:
1855:
1824:
1822:
1821:
1816:
1802:
1801:
1772:
1770:
1769:
1764:
1762:
1761:
1734:
1733:
1705:
1703:
1702:
1697:
1692:
1684:
1683:
1678:
1658:
1656:
1655:
1650:
1575:
1573:
1572:
1567:
1565:
1564:
1504:
1502:
1501:
1496:
1441:
1439:
1438:
1433:
1431:
1430:
1403:
1402:
1383:
1381:
1380:
1375:
1373:
1372:
1300:
1299:
1253:, we find that
1252:
1250:
1249:
1244:
1223:
1221:
1220:
1215:
1197:
1195:
1194:
1189:
1171:
1169:
1168:
1163:
1142:
1140:
1139:
1134:
1132:
1131:
1104:
1103:
1078:
1076:
1075:
1070:
1068:
951:
949:
948:
943:
938:
935:
888:
886:
885:
880:
878:
877:
852:
851:
850:
845:
809:
807:
806:
801:
774:
772:
771:
766:
764:
763:
762:
761:
749:
748:
733:
732:
731:
726:
683:
681:
680:
675:
667:
666:
651:
650:
628:
626:
625:
620:
618:
617:
602:
601:
589:
588:
573:
572:
560:
559:
547:
546:
530:
528:
527:
522:
520:
519:
504:
503:
481:
479:
478:
473:
471:
470:
455:
454:
432:
430:
429:
424:
422:
421:
406:
405:
383:
381:
380:
375:
373:
372:
357:
356:
334:
332:
331:
326:
324:
292:
290:
289:
284:
282:
281:
269:
268:
189:
187:
186:
181:
175:
174:
147:
146:
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
8661:
8660:
8656:
8655:
8654:
8652:
8651:
8650:
8646:Quadratic forms
8636:
8635:
8609:
8560:
8548:. Foreword by
8546:J. H. Silverman
8515:
8478:
8468:Springer-Verlag
8438:
8434:
8425:
8421:
8419:
8416:
8415:
8388:
8383:
8375:
8371:
8363:
8359:
8351:
8347:
8339:
8335:
8327:
8323:
8315:
8311:
8303:
8299:
8291:
8287:
8280:
8276:
8269:
8265:
8257:
8253:
8245:
8241:
8233:
8229:
8225:
8212:Legendre symbol
8198:
8065:in 1775 in his
8020:Pell's equation
8015:
7990:
7968:
7964:
7940:
7936:
7931:
7928:
7927:
7926:since this and
7910:
7906:
7882:
7878:
7873:
7870:
7869:
7852:
7848:
7824:
7820:
7815:
7812:
7811:
7794:
7790:
7766:
7762:
7757:
7754:
7753:
7722:
7714:
7703:
7700:
7699:
7682:
7678:
7654:
7650:
7648:
7645:
7644:
7624:
7621:
7620:
7603:
7599:
7597:
7594:
7593:
7576:
7572:
7570:
7567:
7566:
7549:
7545:
7543:
7540:
7539:
7522:
7518:
7516:
7513:
7512:
7495:
7491:
7489:
7486:
7485:
7468:
7464:
7462:
7459:
7458:
7441:
7437:
7435:
7432:
7431:
7414:
7410:
7408:
7405:
7404:
7380:
7376:
7352:
7348:
7343:
7340:
7339:
7313:
7312:
7307:
7301:
7300:
7295:
7285:
7284:
7282:
7279:
7278:
7241:
7237:
7235:
7232:
7231:
7214:
7210:
7208:
7205:
7204:
7187:
7183:
7159:
7155:
7150:
7147:
7146:
7109:
7105:
7097:
7094:
7093:
7063:
7060:
7059:
7043:
7042:
7023:
7012:
7005:
7001:
6995:
6991:
6984:
6981:
6974:
6959:
6955:
6952:
6949:
6948:
6932:
6928:
6925:
6913:
6907:
6903:
6896:
6890:
6889:
6873:
6869:
6866:
6854:
6848:
6844:
6837:
6830:
6828:
6825:
6824:
6800:
6796:
6789:
6785:
6779:
6775:
6774:
6771:
6763:
6760:
6759:
6739:
6735:
6726:
6722:
6713:
6709:
6695:
6692:
6691:
6667:
6663:
6654:
6650:
6649:
6646:
6637:
6633:
6631:
6628:
6627:
6607:
6604:
6603:
6586:
6582:
6576:
6572:
6557:
6553:
6544:
6540:
6534:
6530:
6521:
6517:
6515:
6512:
6511:
6494:
6490:
6484:
6480:
6465:
6461:
6452:
6448:
6442:
6438:
6429:
6425:
6423:
6420:
6419:
6408:
6374:
6371:
6370:
6350:
6347:
6346:
6345:. For negative
6330:
6327:
6326:
6305:
6297:
6295:
6292:
6291:
6289:quadratic field
6264:
6261:
6260:
6240:
6237:
6236:
6218:
6186:
6185:
6173:
6146:
6143:
6142:
6120:
6117:
6116:
6075:
6072:
6071:
6055:
6052:
6051:
6025:
6022:
6021:
6005:
6002:
6001:
5975:
5972:
5971:
5964:representatives
5947:
5944:
5943:
5915:
5911:
5899:
5895:
5887:
5884:
5883:
5863:
5859:
5847:
5843:
5831:
5827:
5815:
5811:
5794:
5791:
5790:
5770:
5766:
5757:
5753:
5739:
5736:
5735:
5710:
5709:
5704:
5695:
5694:
5686:
5676:
5675:
5673:
5670:
5669:
5646:
5642:
5633:
5629:
5615:
5612:
5611:
5594:
5590:
5578:
5574:
5566:
5563:
5562:
5539:
5530:
5522:
5521:
5519:
5516:
5515:
5494:
5493:
5487:
5483:
5480:
5479:
5473:
5469:
5462:
5461:
5451:
5450:
5444:
5440:
5437:
5436:
5430:
5426:
5419:
5418:
5411:
5410:
5405:
5396:
5395:
5387:
5377:
5376:
5374:
5371:
5370:
5291:
5288:
5287:
5266:
5265:
5260:
5254:
5253:
5248:
5238:
5237:
5235:
5232:
5231:
5208:
5204:
5195:
5191:
5177:
5174:
5173:
5153:
5149:
5140:
5136:
5122:
5119:
5118:
5111:
5105:is indefinite.
5089:
5085:
5073:
5069:
5067:
5064:
5063:
5046:
5042:
5033:
5029:
5027:
5024:
5023:
4974:
4971:
4970:
4948:
4945:
4944:
4927:
4923:
4899:
4895:
4884:
4881:
4880:
4829:
4825:
4823:
4820:
4819:
4781:
4777:
4775:
4772:
4771:
4736:
4732:
4714:
4710:
4686:
4682:
4680:
4677:
4676:
4652:
4648:
4639:
4635:
4633:
4630:
4629:
4599:
4595:
4593:
4590:
4589:
4569:
4565:
4556:
4552:
4550:
4547:
4546:
4529:
4525:
4516:
4512:
4510:
4507:
4506:
4489:
4485:
4473:
4469:
4467:
4464:
4463:
4446:
4442:
4433:
4429:
4427:
4424:
4423:
4406:
4402:
4390:
4386:
4384:
4381:
4380:
4363:
4359:
4350:
4346:
4344:
4341:
4340:
4317:
4314:
4313:
4306:
4280:
4276:
4264:
4260:
4258:
4255:
4254:
4237:
4236:
4227:
4226:
4201:
4147:
4146:
4121:
4066:
4064:
4061:
4060:
4040:
4036:
4024:
4020:
4012:
4009:
4008:
3980:
3977:
3976:
3959:
3955:
3940:
3936:
3928:
3925:
3924:
3838:
3835:
3834:
3803:
3800:
3799:
3747:
3744:
3743:
3726:
3722:
3710:
3706:
3698:
3695:
3694:
3666:
3663:
3662:
3645:
3641:
3626:
3622:
3614:
3611:
3610:
3564:
3561:
3560:
3535:
3531:
3519:
3515:
3507:
3504:
3503:
3492:Pell's equation
3470:
3468:
3465:
3464:
3448:
3440:
3438:
3435:
3434:
3418:
3410:
3408:
3405:
3404:
3388:
3385:
3384:
3367:
3363:
3354:
3350:
3348:
3345:
3344:
3327:
3323:
3314:
3310:
3302:
3299:
3298:
3297:, the equation
3282:
3279:
3278:
3258:
3254:
3245:
3241:
3233:
3230:
3229:
3204:
3201:
3200:
3184:
3181:
3180:
3164:
3161:
3160:
3132:
3129:
3128:
3111:
3107:
3098:
3094:
3092:
3089:
3088:
3072:
3069:
3068:
3037:
3034:
3033:
3010:
3006:
3004:
3001:
3000:
2980:
2976:
2967:
2963:
2955:
2952:
2951:
2947:, the equation
2926:
2923:
2922:
2906:
2903:
2902:
2886:
2883:
2882:
2866:
2863:
2862:
2846:
2843:
2842:
2813: and
2811:
2776:
2773:
2772:
2755:
2754:
2745:
2741:
2732:
2728:
2721:
2715:
2714:
2705:
2701:
2692:
2688:
2681:
2674:
2672:
2669:
2668:
2643:
2640:
2639:
2622:
2618:
2609:
2605:
2597:
2594:
2593:
2577:
2574:
2573:
2557:
2554:
2553:
2537:
2534:
2533:
2527:
2516:
2510:
2466:
2463:
2462:
2446:
2443:
2442:
2426:
2423:
2422:
2406:
2403:
2402:
2368:
2365:
2364:
2361:
2325:
2316:
2308:
2307:
2305:
2302:
2301:
2280:
2276:
2264:
2260:
2252:
2249:
2248:
2227:
2226:
2221:
2212:
2211:
2203:
2193:
2192:
2190:
2187:
2186:
2107:
2104:
2103:
2076:
2067:
2059:
2058:
2056:
2053:
2052:
2031:
2030:
2025:
2019:
2018:
2013:
2003:
2002:
2000:
1997:
1996:
1986:
1952:
1935:
1932:
1931:
1903:
1900:
1899:
1873:
1870:
1869:
1849:
1846:
1845:
1797:
1793:
1785:
1782:
1781:
1757:
1753:
1729:
1725:
1714:
1711:
1710:
1688:
1679:
1671:
1670:
1668:
1665:
1664:
1584:
1581:
1580:
1559:
1558:
1553:
1547:
1546:
1541:
1531:
1530:
1528:
1525:
1524:
1469:
1466:
1465:
1459:class invariant
1426:
1422:
1398:
1394:
1389:
1386:
1385:
1368:
1364:
1295:
1291:
1262:
1259:
1258:
1229:
1226:
1225:
1203:
1200:
1199:
1177:
1174:
1173:
1148:
1145:
1144:
1127:
1123:
1099:
1095:
1087:
1084:
1083:
1066:
1065:
1055:
1037:
1036:
1008:
962:
960:
957:
956:
936: and
934:
914:
911:
910:
895:
858:
854:
846:
841:
840:
821:
815:
812:
811:
780:
777:
776:
757:
753:
744:
740:
739:
735:
727:
722:
721:
702:
696:
693:
692:
662:
658:
646:
642:
640:
637:
636:
634:Pell's equation
613:
609:
597:
593:
584:
580:
568:
564:
555:
551:
542:
538:
536:
533:
532:
515:
511:
499:
495:
487:
484:
483:
466:
462:
450:
446:
438:
435:
434:
417:
413:
401:
397:
389:
386:
385:
368:
364:
352:
348:
340:
337:
336:
309:
298:
295:
294:
277:
273:
264:
260:
252:
249:
248:
211:complex numbers
170:
166:
142:
138:
112:
109:
108:
100:is a quadratic
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
17:
12:
11:
5:
8659:
8649:
8648:
8634:
8633:
8615:
8608:
8607:External links
8605:
8604:
8603:
8591:
8579:
8558:
8526:
8513:
8497:Taylor, Martin
8489:
8476:
8456:
8441:
8437:
8433:
8428:
8424:
8410:
8403:
8387:
8384:
8382:
8381:
8369:
8357:
8345:
8333:
8321:
8309:
8297:
8285:
8274:
8263:
8251:
8239:
8226:
8224:
8221:
8220:
8219:
8214:
8209:
8204:
8197:
8194:
8190:Bhargava cubes
8131:Disquisitiones
8077:infrastructure
8014:
8011:
7989:
7986:
7971:
7967:
7963:
7960:
7957:
7954:
7951:
7948:
7943:
7939:
7935:
7913:
7909:
7905:
7902:
7899:
7896:
7893:
7890:
7885:
7881:
7877:
7855:
7851:
7847:
7844:
7841:
7838:
7835:
7832:
7827:
7823:
7819:
7797:
7793:
7789:
7786:
7783:
7780:
7777:
7774:
7769:
7765:
7761:
7736:
7733:
7729:
7726:
7721:
7716: or
7713:
7710:
7707:
7685:
7681:
7677:
7674:
7671:
7668:
7665:
7662:
7657:
7653:
7628:
7606:
7602:
7579:
7575:
7552:
7548:
7525:
7521:
7498:
7494:
7471:
7467:
7444:
7440:
7417:
7413:
7383:
7379:
7375:
7372:
7369:
7366:
7363:
7360:
7355:
7351:
7347:
7332:
7331:
7317:
7311:
7308:
7306:
7303:
7302:
7299:
7296:
7294:
7291:
7290:
7288:
7244:
7240:
7217:
7213:
7190:
7186:
7182:
7179:
7176:
7173:
7170:
7167:
7162:
7158:
7154:
7143:
7142:
7141:is an integer.
7126:
7123:
7120:
7117:
7112:
7108:
7104:
7101:
7086:
7070:
7067:
7040:
7037:
7034:
7030:
7027:
7018:
7015:
7008:
7004:
6998:
6994:
6990:
6987:
6980:
6977:
6975:
6973:
6967:
6962:
6958:
6951:
6950:
6946:
6940:
6935:
6931:
6924:
6920:
6917:
6910:
6906:
6902:
6899:
6897:
6895:
6892:
6891:
6887:
6881:
6876:
6872:
6865:
6861:
6858:
6851:
6847:
6843:
6840:
6838:
6836:
6833:
6832:
6819:
6803:
6799:
6792:
6788:
6782:
6778:
6770:
6767:
6747:
6742:
6738:
6734:
6729:
6725:
6721:
6716:
6712:
6708:
6705:
6702:
6699:
6676:
6670:
6666:
6662:
6657:
6653:
6645:
6640:
6636:
6611:
6589:
6585:
6579:
6575:
6571:
6568:
6565:
6560:
6556:
6552:
6547:
6543:
6537:
6533:
6529:
6524:
6520:
6497:
6493:
6487:
6483:
6479:
6476:
6473:
6468:
6464:
6460:
6455:
6451:
6445:
6441:
6437:
6432:
6428:
6413:Bhargava cubes
6407:
6404:
6378:
6354:
6334:
6314:
6309:
6304:
6300:
6268:
6244:
6217:
6214:
6172:
6169:
6156:
6153:
6150:
6130:
6127:
6124:
6100:
6097:
6094:
6091:
6088:
6085:
6082:
6079:
6059:
6035:
6032:
6029:
6009:
5985:
5982:
5979:
5951:
5918:
5914:
5910:
5907:
5902:
5898:
5894:
5891:
5871:
5866:
5862:
5858:
5855:
5850:
5846:
5842:
5839:
5834:
5830:
5826:
5823:
5818:
5814:
5810:
5807:
5804:
5801:
5798:
5778:
5773:
5769:
5765:
5760:
5756:
5752:
5749:
5746:
5743:
5728:
5727:
5714:
5708:
5705:
5703:
5700:
5697:
5696:
5693:
5690:
5687:
5685:
5682:
5681:
5679:
5654:
5649:
5645:
5641:
5636:
5632:
5628:
5625:
5622:
5619:
5597:
5593:
5589:
5586:
5581:
5577:
5573:
5570:
5546:
5542:
5538:
5533:
5528:
5525:
5512:
5511:
5498:
5490:
5486:
5482:
5481:
5476:
5472:
5468:
5467:
5465:
5460:
5455:
5447:
5443:
5439:
5438:
5433:
5429:
5425:
5424:
5422:
5415:
5409:
5406:
5404:
5401:
5398:
5397:
5394:
5391:
5388:
5386:
5383:
5382:
5380:
5355:
5352:
5349:
5346:
5343:
5340:
5337:
5334:
5331:
5328:
5325:
5322:
5319:
5316:
5313:
5310:
5307:
5304:
5301:
5298:
5295:
5284:
5283:
5270:
5264:
5261:
5259:
5256:
5255:
5252:
5249:
5247:
5244:
5243:
5241:
5216:
5211:
5207:
5203:
5198:
5194:
5190:
5187:
5184:
5181:
5161:
5156:
5152:
5148:
5143:
5139:
5135:
5132:
5129:
5126:
5110:
5107:
5092:
5088:
5084:
5081:
5076:
5072:
5049:
5045:
5041:
5036:
5032:
4984:
4981:
4978:
4958:
4955:
4952:
4930:
4926:
4922:
4919:
4916:
4913:
4910:
4907:
4902:
4898:
4894:
4891:
4888:
4877:
4876:
4873:
4870:
4843:
4840:
4837:
4832:
4828:
4795:
4792:
4789:
4784:
4780:
4768:
4767:
4756:
4753:
4750:
4747:
4744:
4739:
4735:
4731:
4728:
4725:
4722:
4717:
4713:
4709:
4706:
4703:
4700:
4697:
4694:
4689:
4685:
4655:
4651:
4647:
4642:
4638:
4613:
4610:
4607:
4602:
4598:
4572:
4568:
4564:
4559:
4555:
4532:
4528:
4524:
4519:
4515:
4492:
4488:
4484:
4481:
4476:
4472:
4449:
4445:
4441:
4436:
4432:
4409:
4405:
4401:
4398:
4393:
4389:
4366:
4362:
4358:
4353:
4349:
4321:
4305:
4302:
4283:
4279:
4275:
4272:
4267:
4263:
4251:
4250:
4235:
4232:
4230:
4228:
4225:
4222:
4219:
4216:
4213:
4210:
4207:
4204:
4202:
4200:
4197:
4194:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4158:
4155:
4152:
4149:
4148:
4145:
4142:
4139:
4136:
4133:
4130:
4127:
4124:
4122:
4120:
4117:
4114:
4111:
4108:
4105:
4102:
4099:
4096:
4093:
4090:
4087:
4084:
4081:
4078:
4075:
4072:
4069:
4068:
4043:
4039:
4035:
4032:
4027:
4023:
4019:
4016:
3996:
3993:
3990:
3987:
3984:
3962:
3958:
3954:
3951:
3948:
3943:
3939:
3935:
3932:
3921:
3920:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3819:
3816:
3813:
3810:
3807:
3787:
3784:
3781:
3778:
3775:
3772:
3769:
3766:
3763:
3760:
3757:
3754:
3751:
3729:
3725:
3721:
3718:
3713:
3709:
3705:
3702:
3682:
3679:
3676:
3673:
3670:
3648:
3644:
3640:
3637:
3634:
3629:
3625:
3621:
3618:
3598:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3574:
3571:
3568:
3538:
3534:
3530:
3527:
3522:
3518:
3514:
3511:
3474:
3451:
3447:
3443:
3421:
3417:
3413:
3392:
3370:
3366:
3362:
3357:
3353:
3330:
3326:
3322:
3317:
3313:
3309:
3306:
3286:
3261:
3257:
3253:
3248:
3244:
3240:
3237:
3217:
3214:
3211:
3208:
3199:each equal to
3188:
3168:
3148:
3145:
3142:
3139:
3136:
3114:
3110:
3106:
3101:
3097:
3076:
3056:
3053:
3050:
3047:
3044:
3041:
3021:
3018:
3013:
3009:
2997:
2996:
2983:
2979:
2975:
2970:
2966:
2962:
2959:
2936:
2933:
2930:
2910:
2890:
2870:
2850:
2830:
2827:
2824:
2821:
2818:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2769:
2768:
2753:
2748:
2744:
2740:
2735:
2731:
2727:
2724:
2722:
2720:
2717:
2716:
2713:
2708:
2704:
2700:
2695:
2691:
2687:
2684:
2682:
2680:
2677:
2676:
2653:
2650:
2647:
2625:
2621:
2617:
2612:
2608:
2604:
2601:
2581:
2561:
2541:
2526:
2523:
2507:representation
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2450:
2430:
2410:
2387:
2384:
2381:
2378:
2375:
2372:
2360:
2359:Representation
2357:
2332:
2328:
2324:
2319:
2314:
2311:
2283:
2279:
2275:
2272:
2267:
2263:
2259:
2256:
2245:
2244:
2231:
2225:
2222:
2220:
2217:
2214:
2213:
2210:
2207:
2204:
2202:
2199:
2198:
2196:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2083:
2079:
2075:
2070:
2065:
2062:
2049:
2048:
2035:
2029:
2026:
2024:
2021:
2020:
2017:
2014:
2012:
2009:
2008:
2006:
1985:
1982:
1970:
1966:
1963:
1959:
1956:
1951:
1948:
1945:
1942:
1939:
1907:
1883:
1880:
1877:
1853:
1842:
1841:
1826:
1814:
1811:
1808:
1805:
1800:
1796:
1792:
1789:
1760:
1756:
1752:
1749:
1746:
1743:
1740:
1737:
1732:
1728:
1724:
1721:
1718:
1695:
1691:
1687:
1682:
1677:
1674:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1577:
1576:
1563:
1557:
1554:
1552:
1549:
1548:
1545:
1542:
1540:
1537:
1536:
1534:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1429:
1425:
1421:
1418:
1415:
1412:
1409:
1406:
1401:
1397:
1393:
1371:
1367:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1298:
1294:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1266:
1242:
1239:
1236:
1233:
1213:
1210:
1207:
1187:
1184:
1181:
1161:
1158:
1155:
1152:
1130:
1126:
1122:
1119:
1116:
1113:
1110:
1107:
1102:
1098:
1094:
1091:
1080:
1079:
1064:
1061:
1058:
1056:
1054:
1051:
1048:
1045:
1042:
1039:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1009:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
980:
977:
974:
971:
968:
965:
964:
941:
933:
930:
927:
924:
921:
918:
894:
891:
876:
873:
870:
867:
864:
861:
857:
849:
844:
839:
836:
833:
830:
827:
824:
820:
799:
796:
793:
790:
787:
784:
760:
756:
752:
747:
743:
738:
730:
725:
720:
717:
714:
711:
708:
705:
701:
673:
670:
665:
661:
657:
654:
649:
645:
616:
612:
608:
605:
600:
596:
592:
587:
583:
579:
576:
571:
567:
563:
558:
554:
550:
545:
541:
518:
514:
510:
507:
502:
498:
494:
491:
469:
465:
461:
458:
453:
449:
445:
442:
420:
416:
412:
409:
404:
400:
396:
393:
371:
367:
363:
360:
355:
351:
347:
344:
323:
320:
316:
313:
308:
305:
302:
280:
276:
272:
267:
263:
259:
256:
215:quadratic form
191:
190:
178:
173:
169:
165:
162:
159:
156:
153:
150:
145:
141:
137:
134:
131:
128:
125:
122:
119:
116:
88:
87:
42:
40:
33:
25:quadratic form
15:
9:
6:
4:
3:
2:
8658:
8647:
8644:
8643:
8641:
8631:
8627:
8626:
8621:
8616:
8614:
8611:
8610:
8600:
8596:
8592:
8588:
8584:
8580:
8577:
8573:
8569:
8565:
8561:
8555:
8551:
8547:
8543:
8540:, Revised by
8539:
8535:
8534:Wright, E. M.
8531:
8527:
8524:
8520:
8516:
8510:
8506:
8502:
8498:
8494:
8490:
8487:
8483:
8479:
8473:
8469:
8465:
8461:
8457:
8455:
8439:
8435:
8431:
8426:
8422:
8412:David A Cox,
8411:
8408:
8404:
8402:
8401:3-540-46367-4
8398:
8394:
8390:
8389:
8378:
8373:
8366:
8361:
8354:
8349:
8342:
8337:
8330:
8325:
8318:
8313:
8306:
8301:
8294:
8289:
8283:
8278:
8272:
8267:
8260:
8255:
8248:
8243:
8236:
8231:
8227:
8218:
8215:
8213:
8210:
8208:
8205:
8203:
8202:Bhargava cube
8200:
8199:
8193:
8191:
8187:
8183:
8179:
8175:
8169:
8167:
8163:
8159:
8158:
8154:
8150:
8146:
8142:
8141:
8136:
8132:
8128:
8127:number fields
8122:
8120:
8116:
8112:
8108:
8103:
8099:
8098:
8093:
8088:
8086:
8082:
8078:
8074:
8070:
8069:
8064:
8059:
8057:
8053:
8049:
8045:
8041:
8037:
8033:
8029:
8025:
8021:
8010:
8008:
8004:
8000:
7995:
7985:
7969:
7965:
7961:
7958:
7955:
7952:
7949:
7946:
7941:
7937:
7933:
7911:
7907:
7903:
7900:
7897:
7894:
7891:
7888:
7883:
7879:
7875:
7853:
7849:
7845:
7842:
7839:
7836:
7833:
7830:
7825:
7821:
7817:
7795:
7791:
7787:
7784:
7781:
7778:
7775:
7772:
7767:
7763:
7759:
7751:
7731:
7727:
7719:
7711:
7708:
7683:
7679:
7675:
7672:
7669:
7666:
7663:
7660:
7655:
7651:
7642:
7604:
7600:
7577:
7573:
7550:
7546:
7523:
7519:
7496:
7492:
7469:
7465:
7442:
7438:
7415:
7411:
7401:
7399:
7381:
7377:
7373:
7370:
7367:
7364:
7361:
7358:
7353:
7349:
7345:
7337:
7315:
7309:
7304:
7297:
7292:
7286:
7277:
7276:
7275:
7272:
7268:
7264:
7260:
7242:
7238:
7215:
7211:
7188:
7184:
7180:
7177:
7174:
7171:
7168:
7165:
7160:
7156:
7152:
7140:
7124:
7121:
7118:
7115:
7110:
7106:
7102:
7091:
7087:
7084:
7068:
7065:
7056:
7035:
7032:
7028:
7016:
7013:
7006:
7002:
6996:
6992:
6988:
6978:
6976:
6971:
6965:
6960:
6956:
6938:
6933:
6929:
6922:
6918:
6908:
6904:
6900:
6898:
6893:
6879:
6874:
6870:
6863:
6859:
6849:
6845:
6841:
6839:
6834:
6820:
6801:
6797:
6790:
6786:
6780:
6776:
6768:
6765:
6740:
6736:
6732:
6727:
6723:
6719:
6714:
6710:
6700:
6697:
6674:
6668:
6664:
6660:
6655:
6651:
6643:
6638:
6634:
6625:
6624:
6623:
6587:
6583:
6577:
6573:
6569:
6566:
6563:
6558:
6554:
6550:
6545:
6541:
6535:
6531:
6527:
6522:
6518:
6495:
6491:
6485:
6481:
6477:
6474:
6471:
6466:
6462:
6458:
6453:
6449:
6443:
6439:
6435:
6430:
6426:
6416:
6414:
6403:
6401:
6396:
6393:
6390:
6368:
6290:
6286:
6282:
6258:
6234:
6230:
6229:abelian group
6226:
6222:
6213:
6211:
6210:
6204:
6202:
6198:
6194:
6190:
6182:
6178:
6168:
6154:
6151:
6148:
6128:
6125:
6122:
6114:
6095:
6092:
6089:
6083:
6080:
6077:
6049:
6033:
6030:
5999:
5983:
5980:
5969:
5968:reduced forms
5965:
5941:
5936:
5934:
5916:
5912:
5908:
5905:
5900:
5896:
5892:
5889:
5864:
5860:
5856:
5853:
5848:
5844:
5840:
5837:
5832:
5828:
5824:
5821:
5816:
5812:
5808:
5802:
5799:
5796:
5771:
5767:
5763:
5758:
5754:
5747:
5744:
5741:
5733:
5712:
5706:
5701:
5698:
5691:
5688:
5683:
5677:
5668:
5667:
5666:
5647:
5643:
5639:
5634:
5630:
5623:
5620:
5617:
5595:
5591:
5587:
5584:
5579:
5575:
5571:
5568:
5559:
5531:
5496:
5488:
5484:
5474:
5470:
5463:
5458:
5453:
5445:
5441:
5431:
5427:
5420:
5413:
5407:
5402:
5399:
5392:
5389:
5384:
5378:
5369:
5368:
5367:
5350:
5347:
5344:
5338:
5335:
5329:
5326:
5323:
5320:
5317:
5314:
5311:
5308:
5305:
5302:
5299:
5293:
5268:
5262:
5257:
5250:
5245:
5239:
5230:
5229:
5228:
5209:
5205:
5201:
5196:
5192:
5185:
5182:
5179:
5154:
5150:
5146:
5141:
5137:
5130:
5127:
5124:
5116:
5106:
5090:
5086:
5082:
5079:
5074:
5070:
5047:
5043:
5039:
5034:
5030:
5021:
5017:
5014:is finite if
5013:
5009:
5004:
5002:
4998:
4982:
4979:
4976:
4956:
4953:
4950:
4928:
4924:
4920:
4917:
4914:
4911:
4908:
4905:
4900:
4896:
4892:
4889:
4886:
4874:
4871:
4868:
4864:
4863:
4862:
4859:
4857:
4838:
4830:
4826:
4817:
4813:
4809:
4790:
4782:
4778:
4754:
4745:
4737:
4733:
4729:
4723:
4715:
4711:
4704:
4701:
4695:
4687:
4683:
4675:
4674:
4673:
4671:
4653:
4649:
4645:
4640:
4636:
4627:
4608:
4600:
4596:
4588:
4570:
4566:
4562:
4557:
4553:
4530:
4526:
4522:
4517:
4513:
4490:
4486:
4482:
4479:
4474:
4470:
4447:
4443:
4439:
4434:
4430:
4407:
4403:
4399:
4396:
4391:
4387:
4364:
4360:
4356:
4351:
4347:
4337:
4335:
4319:
4311:
4301:
4299:
4281:
4277:
4273:
4270:
4265:
4261:
4233:
4231:
4223:
4217:
4214:
4211:
4205:
4203:
4195:
4192:
4189:
4186:
4183:
4180:
4177:
4174:
4171:
4168:
4165:
4162:
4159:
4156:
4153:
4143:
4137:
4134:
4131:
4125:
4123:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4059:
4058:
4057:
4041:
4037:
4033:
4030:
4025:
4021:
4017:
4014:
3991:
3988:
3985:
3960:
3956:
3952:
3949:
3946:
3941:
3937:
3933:
3930:
3903:
3900:
3897:
3891:
3885:
3882:
3879:
3876:
3873:
3870:
3867:
3864:
3861:
3858:
3855:
3852:
3849:
3846:
3843:
3833:
3832:
3831:
3830:, we compute
3814:
3811:
3808:
3782:
3779:
3776:
3773:
3770:
3767:
3764:
3761:
3758:
3755:
3752:
3727:
3723:
3719:
3716:
3711:
3707:
3703:
3700:
3677:
3674:
3671:
3646:
3642:
3638:
3635:
3632:
3627:
3623:
3619:
3616:
3593:
3590:
3587:
3581:
3575:
3572:
3569:
3558:
3554:
3536:
3532:
3528:
3525:
3520:
3516:
3512:
3509:
3501:
3497:
3493:
3488:
3472:
3445:
3415:
3390:
3368:
3364:
3360:
3355:
3351:
3328:
3324:
3320:
3315:
3311:
3307:
3304:
3284:
3275:
3259:
3255:
3251:
3246:
3242:
3238:
3235:
3215:
3212:
3209:
3206:
3186:
3166:
3143:
3140:
3137:
3112:
3108:
3104:
3099:
3095:
3074:
3054:
3051:
3048:
3045:
3042:
3039:
3019:
3016:
3011:
3007:
2981:
2977:
2973:
2968:
2964:
2960:
2957:
2950:
2949:
2948:
2934:
2931:
2928:
2908:
2888:
2868:
2848:
2825:
2822:
2819:
2805:
2802:
2799:
2793:
2787:
2784:
2781:
2751:
2746:
2742:
2738:
2733:
2729:
2725:
2723:
2718:
2711:
2706:
2702:
2698:
2693:
2689:
2685:
2683:
2678:
2667:
2666:
2665:
2651:
2648:
2645:
2623:
2619:
2615:
2610:
2606:
2602:
2599:
2579:
2559:
2539:
2531:
2522:
2519:
2513:
2508:
2492:
2486:
2483:
2480:
2474:
2471:
2468:
2448:
2428:
2408:
2400:
2382:
2379:
2376:
2370:
2356:
2354:
2350:
2346:
2317:
2299:
2281:
2277:
2273:
2270:
2265:
2261:
2257:
2254:
2229:
2223:
2218:
2215:
2208:
2205:
2200:
2194:
2185:
2184:
2183:
2166:
2163:
2160:
2154:
2151:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2109:
2101:
2097:
2068:
2033:
2027:
2022:
2015:
2010:
2004:
1995:
1994:
1993:
1991:
1984:Automorphisms
1981:
1968:
1961:
1957:
1949:
1946:
1943:
1940:
1929:
1925:
1921:
1897:
1881:
1878:
1867:
1839:
1835:
1831:
1827:
1812:
1809:
1806:
1803:
1798:
1794:
1790:
1780:
1776:
1775:
1774:
1758:
1754:
1750:
1747:
1744:
1741:
1738:
1735:
1730:
1726:
1722:
1719:
1716:
1707:
1680:
1662:
1659:is a (right)
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1607:
1598:
1595:
1592:
1586:
1561:
1555:
1550:
1543:
1538:
1532:
1523:
1522:
1521:
1519:
1514:
1512:
1508:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1462:
1460:
1456:
1452:
1448:
1443:
1427:
1423:
1419:
1416:
1413:
1410:
1407:
1404:
1399:
1395:
1391:
1369:
1361:
1358:
1355:
1349:
1346:
1340:
1337:
1334:
1325:
1322:
1319:
1316:
1313:
1310:
1304:
1301:
1296:
1288:
1285:
1282:
1279:
1276:
1273:
1267:
1264:
1256:
1240:
1237:
1234:
1231:
1211:
1208:
1205:
1185:
1182:
1179:
1159:
1156:
1153:
1150:
1128:
1124:
1120:
1117:
1114:
1111:
1108:
1105:
1100:
1096:
1092:
1089:
1062:
1059:
1057:
1052:
1049:
1046:
1043:
1040:
1033:
1027:
1024:
1021:
1015:
1012:
1010:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
966:
955:
954:
953:
939:
931:
928:
925:
922:
919:
916:
908:
904:
900:
890:
871:
868:
865:
859:
855:
847:
837:
831:
828:
825:
818:
794:
791:
788:
782:
758:
754:
750:
745:
741:
736:
728:
718:
712:
709:
706:
699:
689:
685:
671:
668:
663:
659:
655:
652:
647:
643:
635:
630:
614:
610:
606:
603:
598:
594:
590:
585:
581:
577:
574:
569:
565:
561:
556:
552:
548:
543:
539:
516:
512:
508:
505:
500:
496:
492:
489:
467:
463:
459:
456:
451:
447:
443:
440:
418:
414:
410:
407:
402:
398:
394:
391:
369:
365:
361:
358:
353:
349:
345:
342:
318:
314:
306:
303:
300:
278:
274:
270:
265:
261:
257:
254:
245:
243:
242:number fields
239:
235:
230:
228:
224:
220:
216:
212:
208:
204:
200:
196:
176:
171:
167:
163:
160:
157:
154:
151:
148:
143:
139:
135:
132:
126:
123:
120:
114:
107:
106:
105:
103:
99:
95:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
26:
22:
8623:
8598:
8586:
8550:Andrew Wiles
8537:
8530:Hardy, G. H.
8500:
8463:
8460:Cohen, Henri
8413:
8406:
8392:
8372:
8360:
8348:
8336:
8324:
8312:
8300:
8295:, Theorem 58
8288:
8277:
8266:
8254:
8249:, p. 30
8242:
8230:
8170:
8155:
8138:
8130:
8123:
8114:
8110:
8109:in place of
8106:
8095:
8089:
8084:
8073:class number
8066:
8060:
8016:
7998:
7993:
7991:
7402:
7397:
7335:
7333:
7270:
7266:
7262:
7258:
7144:
7138:
7089:
7082:
6823:
6417:
6409:
6397:
6394:
6391:
6257:Class groups
6232:
6220:
6219:
6207:
6205:
6201:reduced form
6200:
6196:
6192:
6187:class number
6184:
6180:
6176:
6174:
6112:
6047:
5997:
5967:
5939:
5937:
5932:
5731:
5729:
5560:
5513:
5285:
5114:
5112:
5019:
5015:
5011:
5007:
5005:
5000:
4996:
4878:
4866:
4860:
4855:
4811:
4769:
4669:
4625:
4338:
4333:
4309:
4307:
4252:
3922:
3556:
3552:
3499:
3495:
3489:
3383:will exceed
3276:
2998:
2770:
2528:
2517:
2511:
2506:
2398:
2362:
2348:
2344:
2246:
2099:
2096:automorphism
2095:
2050:
1989:
1987:
1923:
1919:
1895:
1865:
1843:
1837:
1833:
1829:
1779:discriminant
1708:
1661:group action
1578:
1515:
1510:
1506:
1463:
1458:
1454:
1444:
1254:
1081:
906:
902:
898:
896:
690:
686:
631:
246:
231:
226:
222:
207:coefficients
206:
202:
198:
194:
192:
97:
91:
76:
67:
48:
8595:Zagier, Don
8583:Weil, André
8282:Zagier 1981
8271:Zagier 1981
8157:Zahlbericht
8149:ring theory
8079:. In 1798,
8044:Arithmetica
8036:Bhāskara II
8024:Brahmagupta
6231:called the
6221:Composition
6216:Composition
2401:an integer
1451:partitioned
905:are called
893:Equivalence
94:mathematics
62:introducing
8601:, Springer
8576:1159.11001
8386:References
8331:, Ch.I §IX
8319:, Ch.I §IX
8261:, Thm. 278
8235:Cohen 1993
8186:Bhargava's
8119:Eisenstein
8083:published
8040:Diophantus
7092:such that
6281:isomorphic
5010:by a form
2664:, we have
2592:for which
2530:Diophantus
2399:represents
1920:indefinite
1896:degenerate
907:equivalent
897:Two forms
45:references
8630:EMS Press
8536:(2008) ,
8377:Weil 2001
8365:Weil 2001
8353:Weil 2001
8341:Weil 2001
8329:Weil 2001
8317:Weil 2001
8305:Weil 2001
8247:Weil 2001
8153:Hilbert's
8135:Dirichlet
7947:−
7831:−
7709:≡
7706:Δ
7627:Δ
7145:The form
7116:−
7100:Δ
6986:Δ
6979:≡
6961:μ
6901:≡
6842:≡
6741:μ
6639:μ
6610:Δ
6377:Δ
6353:Δ
6333:Δ
6308:Δ
6267:Δ
6243:Δ
6197:reduction
6152:≥
6111:in which
6058:Δ
6028:Δ
6008:Δ
5978:Δ
5950:Δ
5906:−
5699:−
5689:−
5585:−
5408:α
5403:γ
5400:−
5393:β
5390:−
5385:δ
5327:δ
5318:γ
5309:β
5300:α
5263:δ
5258:γ
5251:β
5246:α
5080:−
4816:congruent
4814:that are
4730:−
4480:−
4397:−
4271:−
4234:⋮
4193:⋅
4181:⋅
4169:⋅
4157:⋅
4113:⋅
4101:⋅
4089:⋅
4077:⋅
4031:−
3953:⋅
3947:−
3883:⋅
3871:⋅
3859:⋅
3847:⋅
3717:−
3639:⋅
3633:−
3526:−
3207:−
3087:. Thus,
3046:−
3017:≥
2271:−
2216:−
2206:−
2143:δ
2134:γ
2125:β
2116:α
2028:δ
2023:γ
2016:β
2011:α
1941:≡
1938:Δ
1924:primitive
1906:Δ
1876:Δ
1852:Δ
1804:−
1788:Δ
1641:δ
1632:γ
1623:β
1614:α
1605:↦
1556:δ
1551:γ
1544:β
1539:α
1490:±
1484:γ
1481:β
1478:−
1475:δ
1472:α
1417:−
1392:−
1359:−
1338:−
1311:−
1274:−
1238:−
1232:δ
1206:γ
1180:β
1157:−
1151:α
1053:γ
1050:β
1047:−
1044:δ
1041:α
1000:δ
991:γ
982:β
973:α
940:δ
929:γ
923:β
917:α
838:∈
819:∑
719:∈
700:∑
653:−
604:−
506:−
457:−
304:≡
238:quadratic
70:July 2009
8640:Category
8597:(1981),
8585:(2001),
8499:(1993),
8462:(1993),
8196:See also
8182:Conway's
8178:Zagier's
8174:Shanks's
8145:Dedekind
8081:Legendre
8063:Lagrange
8032:Jayadeva
7641:identity
7088:Compute
6626:Compute
6020:. When
4808:divisors
3502:so that
2525:Examples
2343:. When
2298:subgroup
1866:definite
205:are the
8568:2445243
8523:1215934
8486:1228206
8379:, p.317
8367:, p.318
8013:History
6287:of the
6283:to the
3742:, then
3032:unless
2638:. When
1455:classes
219:integer
58:improve
21:integer
8574:
8566:
8556:
8521:
8511:
8484:
8474:
8399:
8237:, §5.2
8048:Fermat
7750:invert
7334:where
6758:, and
4770:where
2353:cyclic
2094:is an
1836:, and
1518:matrix
1224:, and
193:where
47:, but
8223:Notes
8102:group
8092:Gauss
8056:Euler
7994:genus
4007:with
3661:. If
775:, if
8554:ISBN
8544:and
8509:ISBN
8472:ISBN
8397:ISBN
7592:and
7538:and
7484:and
7430:and
7261:and
7230:and
6690:and
6510:and
6126:>
6031:>
5981:<
5366:and
5172:and
4980:<
4954:>
3555:and
3498:and
3433:and
3179:and
2901:and
2861:and
2572:and
2441:and
1879:<
1777:The
1143:and
901:and
482:and
96:, a
8572:Zbl
8121:).
8034:or
8009:).
7728:mod
7029:mod
6919:mod
6860:mod
6704:gcd
6279:is
6255:.
4810:of
4628:by
4218:408
4212:577
3067:or
2515:by
2509:of
2300:of
2102:if
2098:of
2051:in
1988:If
1958:mod
1898:if
1868:if
1864:is
1709:If
1663:of
1516:In
315:mod
92:In
8642::
8628:,
8622:,
8570:,
8564:MR
8562:,
8532:;
8519:MR
8517:,
8507:,
8495:;
8482:MR
8480:,
8470:,
8192:.
8168:.
8115:xy
8054:.
8046:,
6415:.
6402:.
6141:,
5003:.
4300:.
4196:70
4184:99
4172:70
4160:99
4138:70
4132:99
4116:12
4104:17
4092:12
4080:17
4056::
3957:12
3938:17
3904:12
3898:17
2719:65
2679:65
2652:65
2521:.
2355:.
1894:,
1832:,
1442:.
1198:,
1172:,
1063:1.
684:.
531:.
433:,
384:,
229:.
201:,
197:,
8440:2
8436:y
8432:+
8427:2
8423:x
8111:b
8107:b
8105:2
7970:2
7966:y
7962:C
7959:+
7956:y
7953:x
7950:B
7942:2
7938:x
7934:A
7912:2
7908:y
7904:A
7901:+
7898:y
7895:x
7892:B
7889:+
7884:2
7880:x
7876:C
7854:2
7850:y
7846:C
7843:+
7840:y
7837:x
7834:B
7826:2
7822:x
7818:A
7796:2
7792:y
7788:C
7785:+
7782:y
7779:x
7776:B
7773:+
7768:2
7764:x
7760:A
7735:)
7732:4
7725:(
7720:1
7712:0
7684:2
7680:y
7676:C
7673:+
7670:y
7667:x
7664:B
7661:+
7656:2
7652:x
7605:2
7601:g
7578:1
7574:g
7551:2
7547:f
7524:1
7520:f
7497:2
7493:g
7470:1
7466:g
7443:2
7439:f
7416:1
7412:f
7398:A
7382:2
7378:y
7374:C
7371:+
7368:y
7365:x
7362:B
7359:+
7354:2
7350:x
7346:A
7336:n
7330:,
7316:)
7310:1
7305:0
7298:n
7293:1
7287:(
7271:B
7267:B
7263:C
7259:B
7243:2
7239:f
7216:1
7212:f
7189:2
7185:y
7181:C
7178:+
7175:y
7172:x
7169:B
7166:+
7161:2
7157:x
7153:A
7139:C
7125:C
7122:A
7119:4
7111:2
7107:B
7103:=
7090:C
7085:.
7083:B
7069:A
7066:2
7039:)
7036:A
7033:2
7026:(
7017:e
7014:2
7007:2
7003:B
6997:1
6993:B
6989:+
6972:x
6966:e
6957:B
6945:)
6939:e
6934:2
6930:A
6923:2
6916:(
6909:2
6905:B
6894:x
6886:)
6880:e
6875:1
6871:A
6864:2
6857:(
6850:1
6846:B
6835:x
6802:2
6798:e
6791:2
6787:A
6781:1
6777:A
6769:=
6766:A
6746:)
6737:B
6733:,
6728:2
6724:A
6720:,
6715:1
6711:A
6707:(
6701:=
6698:e
6675:2
6669:2
6665:B
6661:+
6656:1
6652:B
6644:=
6635:B
6588:2
6584:y
6578:2
6574:C
6570:+
6567:y
6564:x
6559:2
6555:B
6551:+
6546:2
6542:x
6536:2
6532:A
6528:=
6523:2
6519:f
6496:2
6492:y
6486:1
6482:C
6478:+
6475:y
6472:x
6467:1
6463:B
6459:+
6454:2
6450:x
6444:1
6440:A
6436:=
6431:1
6427:f
6313:)
6303:(
6299:Q
6193:D
6181:D
6177:D
6155:0
6149:y
6129:0
6123:x
6113:f
6099:)
6096:y
6093:,
6090:x
6087:(
6084:f
6081:=
6078:n
6048:n
6034:0
5998:n
5984:0
5940:n
5933:f
5917:2
5913:y
5909:2
5901:2
5897:x
5893:=
5890:1
5870:)
5865:1
5861:y
5857:3
5854:+
5849:1
5845:x
5841:2
5838:,
5833:1
5829:y
5825:4
5822:+
5817:1
5813:x
5809:3
5806:(
5803:f
5800:=
5797:1
5777:)
5772:1
5768:y
5764:,
5759:1
5755:x
5751:(
5748:f
5745:=
5742:1
5732:f
5713:)
5707:3
5702:2
5692:4
5684:3
5678:(
5653:)
5648:1
5644:y
5640:,
5635:1
5631:x
5627:(
5624:f
5621:=
5618:1
5596:2
5592:y
5588:2
5580:2
5576:x
5572:=
5569:f
5545:)
5541:Z
5537:(
5532:2
5527:L
5524:S
5497:)
5489:2
5485:y
5475:2
5471:x
5464:(
5459:=
5454:)
5446:1
5442:y
5432:1
5428:x
5421:(
5414:)
5379:(
5354:)
5351:y
5348:,
5345:x
5342:(
5339:g
5336:=
5333:)
5330:y
5324:+
5321:x
5315:,
5312:y
5306:+
5303:x
5297:(
5294:f
5269:)
5240:(
5215:)
5210:2
5206:y
5202:,
5197:2
5193:x
5189:(
5186:g
5183:=
5180:n
5160:)
5155:1
5151:y
5147:,
5142:1
5138:x
5134:(
5131:f
5128:=
5125:m
5091:2
5087:y
5083:2
5075:2
5071:x
5048:2
5044:y
5040:+
5035:2
5031:x
5020:f
5016:f
5012:f
5008:n
4983:0
4977:a
4957:0
4951:a
4929:2
4925:y
4921:c
4918:+
4915:y
4912:x
4909:b
4906:+
4901:2
4897:x
4893:a
4890:=
4887:f
4867:n
4856:n
4842:)
4839:n
4836:(
4831:3
4827:d
4812:n
4794:)
4791:n
4788:(
4783:1
4779:d
4755:,
4752:)
4749:)
4746:n
4743:(
4738:3
4734:d
4727:)
4724:n
4721:(
4716:1
4712:d
4708:(
4705:4
4702:=
4699:)
4696:n
4693:(
4688:2
4684:r
4670:n
4654:2
4650:y
4646:+
4641:2
4637:x
4626:n
4612:)
4609:n
4606:(
4601:2
4597:r
4571:2
4567:y
4563:+
4558:2
4554:x
4531:2
4527:y
4523:+
4518:2
4514:x
4491:2
4487:y
4483:2
4475:2
4471:x
4448:2
4444:y
4440:+
4435:2
4431:x
4408:2
4404:y
4400:2
4392:2
4388:x
4365:2
4361:y
4357:+
4352:2
4348:x
4334:f
4320:n
4282:2
4278:y
4274:2
4266:2
4262:x
4224:,
4221:)
4215:,
4209:(
4206:=
4199:)
4190:3
4187:+
4178:2
4175:,
4166:4
4163:+
4154:3
4151:(
4144:,
4141:)
4135:,
4129:(
4126:=
4119:)
4110:3
4107:+
4098:2
4095:,
4086:4
4083:+
4074:3
4071:(
4042:2
4038:y
4034:2
4026:2
4022:x
4018:=
4015:1
3995:)
3992:y
3989:,
3986:x
3983:(
3961:2
3950:2
3942:2
3934:=
3931:1
3919:,
3907:)
3901:,
3895:(
3892:=
3889:)
3886:2
3880:3
3877:+
3874:3
3868:2
3865:,
3862:2
3856:4
3853:+
3850:3
3844:3
3841:(
3818:)
3815:2
3812:,
3809:3
3806:(
3786:)
3783:y
3780:3
3777:+
3774:x
3771:2
3768:,
3765:y
3762:4
3759:+
3756:x
3753:3
3750:(
3728:2
3724:y
3720:2
3712:2
3708:x
3704:=
3701:1
3681:)
3678:y
3675:,
3672:x
3669:(
3647:2
3643:2
3636:2
3628:2
3624:3
3620:=
3617:1
3597:)
3594:2
3591:,
3588:3
3585:(
3582:=
3579:)
3576:y
3573:,
3570:x
3567:(
3557:y
3553:x
3537:2
3533:y
3529:2
3521:2
3517:x
3513:=
3510:1
3500:y
3496:x
3473:n
3450:|
3446:y
3442:|
3420:|
3416:x
3412:|
3391:n
3369:2
3365:y
3361:+
3356:2
3352:x
3329:2
3325:y
3321:+
3316:2
3312:x
3308:=
3305:n
3285:n
3260:2
3256:y
3252:+
3247:2
3243:x
3239:=
3236:3
3216:0
3213:,
3210:1
3187:y
3167:x
3147:)
3144:y
3141:,
3138:x
3135:(
3113:2
3109:y
3105:+
3100:2
3096:x
3075:1
3055:0
3052:,
3049:1
3043:=
3040:x
3020:4
3012:2
3008:x
2982:2
2978:y
2974:+
2969:2
2965:x
2961:=
2958:3
2935:3
2932:=
2929:n
2909:y
2889:x
2869:y
2849:x
2829:)
2826:7
2823:,
2820:4
2817:(
2809:)
2806:8
2803:,
2800:1
2797:(
2794:=
2791:)
2788:y
2785:,
2782:x
2779:(
2752:,
2747:2
2743:7
2739:+
2734:2
2730:4
2726:=
2712:,
2707:2
2703:8
2699:+
2694:2
2690:1
2686:=
2649:=
2646:n
2624:2
2620:y
2616:+
2611:2
2607:x
2603:=
2600:n
2580:y
2560:x
2540:n
2518:q
2512:n
2493:.
2490:)
2487:y
2484:,
2481:x
2478:(
2475:q
2472:=
2469:n
2449:y
2429:x
2409:n
2386:)
2383:y
2380:,
2377:x
2374:(
2371:q
2349:f
2345:f
2331:)
2327:Z
2323:(
2318:2
2313:L
2310:S
2282:2
2278:y
2274:2
2266:2
2262:x
2258:=
2255:f
2230:)
2224:3
2219:2
2209:4
2201:3
2195:(
2170:)
2167:y
2164:,
2161:x
2158:(
2155:f
2152:=
2149:)
2146:y
2140:+
2137:x
2131:,
2128:y
2122:+
2119:x
2113:(
2110:f
2100:f
2082:)
2078:Z
2074:(
2069:2
2064:L
2061:S
2034:)
2005:(
1990:f
1969:.
1965:)
1962:4
1955:(
1950:1
1947:,
1944:0
1882:0
1840:.
1838:c
1834:b
1830:a
1825:.
1813:c
1810:a
1807:4
1799:2
1795:b
1791:=
1759:2
1755:y
1751:c
1748:+
1745:y
1742:x
1739:b
1736:+
1731:2
1727:x
1723:a
1720:=
1717:f
1694:)
1690:Z
1686:(
1681:2
1676:L
1673:S
1647:)
1644:y
1638:+
1635:x
1629:,
1626:y
1620:+
1617:x
1611:(
1608:f
1602:)
1599:y
1596:,
1593:x
1590:(
1587:f
1562:)
1533:(
1493:1
1487:=
1428:2
1424:y
1420:2
1414:y
1411:x
1408:4
1405:+
1400:2
1396:x
1370:2
1366:)
1362:y
1356:x
1353:(
1350:2
1347:+
1344:)
1341:y
1335:x
1332:(
1329:)
1326:y
1323:2
1320:+
1317:x
1314:3
1308:(
1305:4
1302:+
1297:2
1293:)
1289:y
1286:2
1283:+
1280:x
1277:3
1271:(
1268:=
1265:g
1255:f
1241:1
1235:=
1212:1
1209:=
1186:2
1183:=
1160:3
1154:=
1129:2
1125:y
1121:2
1118:+
1115:y
1112:x
1109:4
1106:+
1101:2
1097:x
1093:=
1090:f
1060:=
1034:,
1031:)
1028:y
1025:,
1022:x
1019:(
1016:g
1013:=
1006:)
1003:y
997:+
994:x
988:,
985:y
979:+
976:x
970:(
967:f
932:,
926:,
920:,
903:g
899:f
875:)
872:n
869:,
866:m
863:(
860:f
856:q
848:2
843:Z
835:)
832:n
829:,
826:m
823:(
798:)
795:y
792:,
789:x
786:(
783:f
759:2
755:n
751:+
746:2
742:m
737:q
729:2
724:Z
716:)
713:n
710:,
707:m
704:(
672:1
669:=
664:2
660:y
656:n
648:2
644:x
615:2
611:y
607:3
599:2
595:x
591:,
586:2
582:y
578:2
575:+
570:2
566:x
562:,
557:2
553:y
549:+
544:2
540:x
517:2
513:y
509:3
501:2
497:x
493:=
490:p
468:2
464:y
460:2
452:2
448:x
444:=
441:p
419:2
415:y
411:3
408:+
403:2
399:x
395:=
392:p
370:2
366:y
362:2
359:+
354:2
350:x
346:=
343:p
322:)
319:4
312:(
307:1
301:p
279:2
275:y
271:+
266:2
262:x
258:=
255:p
203:c
199:b
195:a
177:,
172:2
168:y
164:c
161:+
158:y
155:x
152:b
149:+
144:2
140:x
136:a
133:=
130:)
127:y
124:,
121:x
118:(
115:q
83:)
77:(
72:)
68:(
54:.
27:.
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