3756:
4322:
1005:
2144:
1083:
were developed later by Frénicle de Bessy, Euler, Kausler, Barlow, Legendre, Schopis, Terquem, Bertrand, Lebesgue, Pepin, Tafelmacher, Hilbert, Bendz, Gambioli, Kronecker, Bang, Sommer, Bottari, Rychlik, Nutzhorn, Carmichael, Hancock, Vrǎnceanu, Grant and
Perella, Barbara, and Dolan. For one proof by
3768:
4177:
cannot equal 0 modulo 5, and must equal one of four possibilities: 1, −1, 2, or −2. If they were all different, two would be opposites and their sum modulo 5 would be zero (implying contrary to the assumption of this case that the other one would be 0 modulo 5).
706:. The multiplication of two odd numbers is always odd, but the multiplication of an even number with any number is always even. An odd number raised to a power is always odd and an even number raised to power is always even, so for example
6923:
Discusses various material which is related to the proof of Fermat's Last
Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey's construction, and the conjectures of Serre and of
321:
solves the second. Conversely, any solution of the second equation corresponds to a solution to the first. The second equation is sometimes useful because it makes the symmetry between the three variables
694:
The addition, subtraction and multiplication of even and odd integers obey simple rules. The addition or subtraction of two even numbers or of two odd numbers always produces an even number, e.g.,
1636:). As before, there must be a lower bound on the size of solutions, while this argument always produces a smaller solution than any given one, and thus the original solution is impossible.
46:
larger than 2. In the centuries following the initial statement of the result and before its general proof, various proofs were devised for particular values of the exponent
2114:
would produce another solution, still smaller, and so on. But this is impossible, since natural numbers cannot be shrunk indefinitely. Therefore, the original solution
895:
As described below, however, some number systems do not have unique factorization. This fact led to the failure of Lamé's 1847 general proof of Fermat's Last
Theorem.
6977:
913:, Fermat's Last Theorem has been separated into two cases that are proven separately. The first case (case I) is to show that there are no primitive solutions
805:
states that any natural number can be written in only one way (uniquely) as the product of prime numbers. For example, 42 equals the product of prime numbers
4853:
5187:
Fermat P. "Ad
Problema XX commentarii in ultimam questionem Arithmeticorum Diophanti. Area trianguli rectanguli in numeris non potest esse quadratus",
793:
are odd. The remaining addend is either even or odd; thus, the parities of the values in the sum are either (odd + even = odd) or (odd + odd = even).
691:. If two numbers are both even or both odd, they have the same parity. By contrast, if one is even and the other odd, they have different parity.
6188:
1040:
has no primitive solutions in integers (no pairwise coprime solutions). In turn, this is sufficient to prove Fermat's Last
Theorem for the case
4482:, respectively. Nevertheless, the reasoning of these even-exponent proofs differs from their odd-exponent counterparts. Dirichlet's proof for
4332:
1109:
Fermat's proof demonstrates that no right triangle with integer sides can have an area that is a square. Let the right triangle have sides
1085:
2472:
are pairwise coprime and not all positive. One of the three must be even, whereas the other two are odd. Without loss of generality,
4978:
Bottari A. "Soluzione intere dell'equazione pitagorica e applicazione alla dimostrazione di alcune teoremi dellla teoria dei numeri".
667:. Since every solution to Fermat's equation can be reduced to a primitive solution by dividing by their greatest common divisor
679:
Integers can be divided into even and odd, those that are evenly divisible by two and those that are not. The even integers are
1016:
to show that the area of a right triangle with integer sides can never equal the square of an integer. This result is known as
3044:
as coprime. Since the three factors on the right-hand side are coprime, they must individually equal cubes of smaller integers
6863:
6840:
6818:
6754:
6697:
6686:
6667:
6617:
6593:
6557:
6526:
6483:
7013:
The title of one edition of the PBS television series NOVA, discusses Andrew Wiles's effort to prove Fermat's Last
Theorem.
5693:
2274:
6033:
5279:
1017:
7062:
6983:
6641:
802:
6929:
4691:
6711:
6498:
1455:
are coprime (this can be assumed because otherwise the factors could be cancelled), the greatest common divisor of
3914:
is divided into the two cases (cases II(i) and II(ii)) by
Dirichlet in 1825. Case II(i) is the case which one of
4595:
Kausler CF (1802). "Nova demonstratio theorematis nec summam, nec differentiam duorum cuborum cubum esse posse".
421:
If the right-hand side of the equation is divisible by 13, then the left-hand side is also divisible by 13. Let
5373:(1823). "Recherches sur quelques objets d'analyse indéterminée, et particulièrement sur le théorème de Fermat".
6545:
6405:
4440:
4023:
3826:
3760:
2191:
was published in 1770. Independent proofs were published by several other mathematicians, including
Kausler,
4614:
4373:
4065:
3898:
3838:
3316:
are also coprime. Therefore, since their product is a cube, they are each the cube of smaller integers,
904:
5061:
Carmichael RD (1913). "On the impossibility of certain
Diophantine equations and systems of equations".
4513:
6552:. Graduate Texts in Mathematics. Vol. 50 (3rd printing 2000 ed.). New York: Springer-Verlag.
4076:
7017:
2180:. Euler had a complete and pure elementary proof in 1760, but the result was not published. Later,
6995:
5556:
2208:
19:
4380:
in 1864, 1874 and 1876. Alternative proofs were developed by Théophile Pépin and Edmond
Maillet.
7057:
5579:
4461:. Strictly speaking, these proofs are unnecessary, since these cases follow from the proofs for
3850:
2212:
428:
6508:
6493:
2151:
7030:
6799:
5191:, vol. I, p. 340 (Latin), vol. III, pp. 271–272 (French). Paris:Gauthier-Villars, 1891, 1896.
763:
cannot all be even, for then they would not be coprime; they could all be divided by two. If
6702:
5370:
4635:
4036:
3830:
3772:
2192:
809:, and no other product of prime numbers equals 42, aside from trivial rearrangements such as
7035:
Simon Singh and John Lynch's film tells the enthralling and emotional story of Andrew Wiles.
4701:
6910:
A blog that covers the history of Fermat's Last Theorem from Pierre Fermat to Andrew Wiles.
6536:
6154:
6078:
5735:
5324:
5174:
3834:
1475:
is either 2 (case A) or 1 (case B). The theorem is proven separately for these two cases.
688:
6735:
4159:
to be equivalent modulo 5, which can be seen as follows: Since they are indivisible by 5,
1012:
Only one mathematical proof by Fermat has survived, in which Fermat uses the technique of
702:. Conversely, the addition or subtraction of an odd and even number is always odd, e.g.,
671:, Fermat's Last Theorem can be proven by demonstrating that no primitive solutions exist.
8:
6512:
4193:
can be designated as the two equivalent numbers modulo 5. That equivalence implies that
1146:
6961:
6571:
6473:
5560:
6852:
6723:
6631:
6608:
6518:
6387:
6338:
6261:: Una demonstración nueva del teorema de Fermat para el caso de las sestas potencias".
6094:
5461:
5328:
5078:
2360:
2200:
2170:
1887:
139:
6958:
6859:
6836:
6814:
6783:
6750:
6742:
6727:
6682:
6663:
6637:
6613:
6589:
6553:
6522:
6479:
6098:
5332:
52:. Several of these proofs are described below, including Fermat's proof in the case
6731:
6715:
6379:
6330:
6086:
5920:
5477:
5312:
5070:
4580:, ser. I, "Commentationes Arithmeticae", vol. I, pp. 38–58, Leipzig:Teubner (1915).
3723:
3107:
2420:
2296:
1013:
72:
60:
27:
6915:
6768:
6422:, vol. I, pp. 189–194, Berlin:G. Reimer (1889); reprinted New York:Chelsea (1969).
5874:
5383:
Reprinted in 1825 as the "Second Supplément" for a printing of the 2nd edition of
5000:
4695:
3846:
2204:
6828:
6581:
6532:
6120:
6105:
6057:
5998:
5753:
5320:
4377:
3964:. After Legendre's proof, Dirichlet completed the proof for the case II(ii) for
3755:
117:
6881:
6302:
Kapferer H (1913). "Beweis des Fermatschen Satzes für die Exponenten 6 und 10".
5964:
5949:
5787:
5431:
4369:
3842:
2196:
6795:
6359:
5261:
4563:
4007:
3868:
2365:
2181:
2166:
2147:
910:
6475:
Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem
6123:(1876). "Généralisation du théorème de Lamé sur l'impossibilité de l'équation
5481:
5316:
7051:
6905:
6787:
6764:
6567:
6432:
6408:(1832). "Démonstration du théorème de Fermat pour le cas des 14 puissances".
5904:
4848:
4444:
3854:
1020:. As shown below, his proof is equivalent to demonstrating that the equation
814:
23:
7039:
5582:(1915). "Quelques formes quadratiques et quelques équations indéterminées".
1644:
In this case, the two factors are coprime. Since their product is a square
6627:
5407:
Tentativo per dimostrare il teorema di Fermat sull'equazione indeterminata
4897:
Gambioli D (1901). "Memoria bibliographica sull'ultimo teorema di Fermat".
4669:
2584:
are coprime and have different parity (one is even, the other odd). Since
31:
6949:
6933:
6603:
2268:
6698:"Über Eulers Beweis des großen Fermatschen Satzes für den Exponenten 3."
6550:
Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory
4321:
7002:
6719:
6469:
6391:
6342:
6090:
5082:
2263:
2173:
on 4 August 1753 in which claimed to have a proof of the case in which
1363:
But as Fermat proved, there can be no integer solution to the equation
687:. The property of whether an integer is even (or not) is known as its
5790:(1847). "Mémoire sur la résolution en nombres complexes de l'équation
7044:
Podcast of BBC by Melvin Bragg and several outstanding mathematicians
6966:
6222:
5925:
5908:
5601:
4436:
2216:
6383:
6334:
6061:
5465:
5074:
3950:
is divided by 2. In July 1825, Dirichlet proved the case II(i) for
3926:
is divided by either 5 and 2. Case II(ii) is the case which one of
813:. This unique factorization property is the basis on which much of
6503:. New York: Chelsea Publishing. pp. 545–550, 615–621, 731–776.
6225:(1896). "Über die Auflösbarkeit einiger unbestimmter Gleichungen".
5138:
Grant, Mike, and Perella, Malcolm, "Descending to the irrational",
1004:
820:
One consequence of this unique factorization property is that if a
38:
6956:
5967:(1840). "Mémoire d'analyse indéterminée démontrant que l'équation
6001:(1840). "Démonstration de l'impossibilité de résoudre l'équation
4372:
in 1839. His rather complicated proof was simplified in 1840 by
2143:
1414:
The first step of Fermat's proof is to factor the left-hand side
512:
35:
5821:
Gambioli D (1903–1904). "Intorno all'ultimo teorema di Fermat".
4566:(1738). "Theorematum quorundam arithmeticorum demonstrationes".
2158:
Fermat sent the letters in which he mentioned the case in which
6514:
History of the theory of numbers. Vol. II: Diophantine analysis
5742:(2nd ed.). Königl. Ges. Wiss. Göttingen. pp. 387–391.
4621:. St. Paul's Church-Yard, London: J. Johnson. pp. 144–145.
4439:, Tafelmacher, Lind, Kapferer, Swift, and Breusch. Similarly,
3377:, it too can be expressed in terms of smaller coprime numbers,
1198:
then by algebraic manipulations it would also be the case that
955:. The second case (case II) corresponds to the condition that
6769:"The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles"
5129:, vol. 4, pp. 202–205, București:Edit. Acad. Rep. Soc. Romana.
4715:
Lebesgue VA (1853). "Résolution des équations biquadratiques
4407:
Fermat's Last Theorem has also been proven for the exponents
3825:
This was proven neither independently nor collaboratively by
2535:
are both odd, their sum and difference are both even numbers
6108:(1874). "Sur l'impossibilité de quelques égalités doubles".
138:
equal to 2, the equation has infinitely many solutions, the
6882:"Tables of Fermat "near-misses" – approximate solutions of
5522:
Krey H (1909). "Neuer Beweis eines arithmetischen Satzes".
4879:Öfver diophantiska ekvationen x + y = z
3767:
2744:
and that the two factors are cubes of two smaller numbers,
2083:
is another solution to the original equation, but smaller (
1723:
cannot both be even. Therefore, the sum and difference of
3957:. In September 1825, Legendre proved the case II(ii) for
3867:
is divided into the two cases (cases I and II) defined by
298:
corresponds to a general integer solution to the equation
7007:
6982:, MacTutor History of Mathematical Topics, archived from
3871:. In case I, the exponent 5 does not divide the product
3648:; therefore, they are each the cube of smaller integers,
1517:
form a primitive Pythagorean triple, they can be written
401:
is also divisible by 13. This follows from the equation
5756:(1843). "Théorèmes nouveaux sur l'équation indéterminée
7026:
magazine, describing Andrew Wiles's successful journey.
6804:(3rd ed.), London: Longman, pp. 399, 401–402
5574:
5572:
5111:
Vrǎnceanu G (1966). "Asupra teorema lui Fermat pentru
5098:
Foundations of the Theory of Algebraic Numbers, vol. I
2738:
are coprime. This implies that three does not divide
1844:, exactly one of them is even. For illustration, let
1163:. If the area were equal to the square of an integer
6975:
6662:. New York: The Mathematical Association of America.
6362:(1960). "A simple proof of Fermat's last theorem for
5952:(1839). "Mémoire sur le dernier théorème de Fermat".
5303:
J. J. Mačys (2007). "On Euler's hypothetical proof".
5233:
4286:
Combining the two results and dividing both sides by
3366:
is odd and its cube is equal to a number of the form
4854:
Jahresbericht der Deutschen Mathematiker-Vereinigung
4851:(1897). "Die Theorie der algebraischen Zahlkörper".
4672:(1846). "Théorèmes sur les puissances des nombres".
4489:
was published in 1832, before Lamé's 1839 proof for
262:, it suffices to prove that it has no solutions for
7031:"Documentary Movie on Fermat's Last Theorem (1996)"
5569:
4786:Pepin T (1883). "Étude sur l'équation indéterminée
3833:around 1825. Alternative proofs were developed by
1931:they can be expressed in terms of smaller integers
34:in 1995. The statement of the theorem involves an
6851:
6280:Lind B (1909). "Einige zahlentheoretische Sätze".
5495:Günther S (1878). "Über die unbestimmte Gleichung
4454:, while Kapferer and Breusch each proved the case
2823:, then it can be written in terms of two integers
284:, every positive-integer solution of the equation
6227:Det Kongel. Norske Videnskabers Selskabs Skrifter
5578:
5151:Barbara, Roy, "Fermat's last theorem in the case
1735:are likewise even numbers, so we define integers
1104:
7049:
7022:Edited version of 2,000-word essay published in
6976:O'Connor, John J.; Robertson, Edmund F. (1996),
6297:
6295:
5740:Zur Theorie der complexen Zahlen, Werke, vol. II
5738:(1875). "Neue Theorie der Zerlegung der Cuben".
5243:
5241:
5229:
5227:
5225:
5223:
5221:
4892:
4890:
4888:
4767:. Paris: Leiber et Faraguet. pp. 83–84, 89.
4619:An Elementary Investigation of Theory of Numbers
515:. In other words, the greatest common divisor (
6849:
6354:
6352:
6248:
5820:
5642:(1977), Oslo:Universitetsforlaget, pp. 555–559.
5616:er unmulig i hele tal fra nul forskjellige tal
5387:, Courcier (Paris). Also reprinted in 1909 in
5208:
5206:
4995:
4993:
4816:
4630:
4628:
569:is a solution of Fermat's equation, then so is
251:Therefore, to prove that Fermat's equation has
16:Partial results found before the complete proof
6950:"The bluffer's guide to Fermat's Last Theorem"
6903:
6239:, pp. 19–30, Oslo:Universitetsforlaget (1977).
4922:Reprinted by New York:Springer-Verlag in 1978.
4881:. Uppsala: Almqvist & Wiksells Boktrycken.
4590:
4588:
4586:
4141:This equation forces two of the three numbers
4075:. A more methodical proof is as follows. By
2015:are coprime, they must be squares themselves,
826:th power of a number equals a product such as
6404:
6292:
5839:
5555:
5363:
5344:
5342:
5257:
5255:
5238:
5218:
5060:
4885:
4376:, and still simpler proofs were published by
3134:In this case, the greatest common divisor of
66:
59:, which is an early example of the method of
6679:The Moment of Proof: Mathematical Epiphanies
6349:
5203:
4990:
4690:
4625:
4383:
3971:by the extended argument for the case II(i).
2706:are coprime, the greatest common divisor of
1850:be even; then the numbers may be written as
1337:Multiplying these equations together yields
6927:
6879:
6827:
6580:
6544:
6431:
6321:Swift E (1914). "Solution to Problem 206".
6205:Assoc. Française Avanc. Sci., Saint-Étienne
5997:
5903:
5752:
5540:
5404:
5369:
5302:
5181:
5110:
4932:Bang A (1905). "Nyt Bevis for at Ligningen
4915:
4817:Tafelmacher WLA (1893). "Sobre la ecuación
4770:
4762:
4714:
4642:(3rd ed.). Paris: Firmin Didot Frères.
4634:
4583:
3292:are coprime, and because 3 does not divide
2676:is always an odd number. Therefore, since
854:are coprime (share no prime factors), then
172:leads to a solution for all the factors of
71:Fermat's Last Theorem states that no three
6916:"Galois representations and modular forms"
6913:
6763:
6566:
6492:
6301:
6119:
6104:
6056:
5339:
5252:
5030:Nutzhorn F (1912). "Den ubestemte Ligning
5029:
4896:
4775:. Paris: Mallet-Bachelier. pp. 71–73.
4656:Einige Sätze aus der unbestimmten Analytik
4644:Reprinted in 1955 by A. Blanchard (Paris).
4594:
4543:Traité des Triangles Rectangles en Nombres
370:, then all three numbers are divisible by
6358:
6187:
5942:
5924:
5873:
5651:
5543:Beitrag zum Beweis des Fermatschen Satzes
5494:
5095:
4999:
4847:
4668:
4556:
3026:were divisible by 3, then 3 would divide
6808:
6695:
6576:. Cambridge: Cambridge University Press.
5734:
5247:
4962:
4876:
4613:
3766:
3754:
2809:is odd and if it satisfies an equation
2493:are both odd, they cannot be equal. If
2142:
1003:
6945:The story, the history and the mystery.
6741:
6626:
6602:
6573:Three Lectures on Fermat's Last Theorem
6507:
6320:
6153:
5840:Werebrusow AS (1905). "On the equation
5460:
5212:
4785:
4653:
4562:
1834:; only one of them can be even. Since
1377:, of which this is a special case with
991:divides only one of the three numbers.
145:
7050:
6676:
6279:
6221:
5963:
5948:
5877:(1910). "On Fermat's last theorem for
5786:
5666:Ciencias Fis. Mat. Naturales (Caracas)
5600:
5521:
5430:
5003:(1910). "On Fermat's last theorem for
4931:
4918:Vorlesungen über Zahlentheorie, vol. I
3786:states that no three coprime integers
3621:are coprime, so are the three factors
796:
343:
120:and has a solution for every possible
6957:
6794:
6468:
6249:Tafelmacher WLA (1897). "La ecuación
6191:(1897). "Sur l'équation indéterminée
6157:(1876). "Impossibilité de l'équation
6031:
5691:
5652:Duarte FJ (1944). "Sobre la ecuación
5348:
5277:
4977:
4944:, ikke kan have rationale Løsinger".
4773:Introduction à la Théorie des Nombres
4545:, vol. I, 1676, Paris. Reprinted in
4511:
4312:
4052:
3775:(the only surviving portrait of him).
3215:. Therefore, neither 3 nor 4 divide
3129:
2726:
6833:13 Lectures on Fermat's Last Theorem
6657:
6013: = 0 en nombres entiers".
5979:est impossible en nombres entiers".
5909:"Sur une question de V. A. Lebesgue"
5434:(1865). "Étude des binômes cubiques
5375:Mém. Acad. Roy. Sci. Institut France
4316:
3976:Chronological table of the proof of
2723:is either 1 (case A) or 3 (case B).
2454:, where the three non-zero integers
2225:Chronological table of the proof of
1086:Infinite descent#Non-solvability of
6850:van der Poorten, Alf (1996-03-06).
5604:(1917). "Et bevis for at ligningen
4920:. Leipzig: Teubner. pp. 35–38.
3938:is divided by 5 and another one of
3151:is 3. That implies that 3 divides
2098:). Applying the same procedure to
1639:
1478:
13:
6651:
6586:Fermat's Last Theorem for Amateurs
6034:"Fermat's Last Theorem: Proof for
5694:"Fermat's Last Theorem: Proof for
5280:"Fermat's Last Theorem: Proof for
5266:Vollständige Anleitung zur Algebra
3014:are coprime since 3 cannot divide
2731:In this case, the two factors of
2341:Fermat's son Samuel published the
1076:. Alternative proofs of the case
781:would be even, so at least one of
364:can be divided by a fourth number
14:
7074:
6873:
6813:. American Mathematical Society.
6501:. Volume II. Diophantine Analysis
5509:Sitzungsberichte Böhm. Ges. Wiss.
5268:, Roy.Acad. Sci., St. Petersburg.
5168:Dolan, Stan, "Fermat's method of
3722:. Therefore, by the argument of
3106:. Therefore, by the argument of
2803:. A crucial lemma shows that if
2381:complete and pure elemental proof
803:fundamental theorem of arithmetic
6809:Mozzochi, Charles (2000-12-07).
6677:Benson, Donald C. (2001-04-05).
6633:An Introduction to Number Theory
6499:History of the Theory of Numbers
5385:Essai sur la Théorie des Nombres
4435:have been published by Kausler,
4320:
3707:which yields a smaller solution
3091:which yields a smaller solution
2423:. The proof assumes a solution
2398:incomplete but elegant proof in
1609:which produces another solution
717:Consider any primitive solution
317:solves the first equation, then
6425:
6398:
6314:
6273:
6242:
6215:
6181:
6147:
6050:
6025:
5991:
5933:
5897:
5867:
5833:
5814:
5780:
5746:
5728:
5719:
5710:
5685:
5676:
5645:
5594:
5549:
5534:
5515:
5488:
5454:
5424:
5398:
5354:
5296:
5271:
5234:O'Connor & Robertson (1996)
5194:
5162:
5145:
5132:
5104:
5089:
5054:
5023:
4971:
4956:
4925:
4909:
4870:
4865:Gesammelte Abhandlungen, vol. I
4841:
4810:
4779:
4708:
4684:
4640:Théorie des Nombres (Volume II)
3409:A short calculation shows that
3167:in terms of a smaller integer,
3032:, violating the designation of
1018:Fermat's right triangle theorem
872:th power of two other numbers,
674:
222:is a solution for the exponent
6854:Notes on Fermat's Last Theorem
6747:Fermat and the Missing Numbers
4965:Vorlesungen über Zahlentheorie
4662:
4647:
4607:
4535:
4526:
4505:
3911:
3901:(1823) if the auxiliary prime
3887:
3761:Peter Gustav Lejeune Dirichlet
2419:, Euler used the technique of
1105:Application to right triangles
116:equal to 1, the equation is a
1:
6835:. New York: Springer Verlag.
6658:Bell, Eric T. (1998-08-06) .
6588:. New York: Springer-Verlag.
6462:
5913:Annales de l'Institut Fourier
4765:Exercices d'Analyse Numérique
4064:can be proven immediately by
3897:can be proven immediately by
3877:. In case II, 5 does divide
1650:, they must each be a square
1008:Portrait of Pierre de Fermat.
683:whereas the odd integers are
6906:"Fermat's Last Theorem Blog"
6478:. Four Walls Eight Windows.
6237:Selected Mathematical Papers
5640:Selected Mathematical Papers
4700:. Paris: Hachette. pp.
4697:Traité Élémentaire d'Algèbre
4181:Without loss of generality,
3472:is odd. The expression for
2922:must be coprime, too. Since
2572:where the non-zero integers
1711:, an even number, and since
949:does not divide the product
898:
647:A pairwise coprime solution
348:If two of the three numbers
7:
6681:. Oxford University Press.
2478:may be assumed to be even.
2412:As Fermat did for the case
2203:, Günther, Gambioli, Krey,
2035:. This gives the equation
10:
7079:
6612:. New York: Anchor Books.
6263:Ann. Univ. Chile, Santiago
5142:83, July 1999, pp.263-267.
3779:Fermat's Last Theorem for
3585:is an integer that equals
3360:By the lemma above, since
2520:is even, a contradiction.
1292:which can be expressed as
902:
519:) of each pair equals one
278:For any such odd exponent
67:Mathematical preliminaries
5482:10.1017/S0370164600041857
5360:Ribenboim, pp. 33, 37–41.
5317:10.1134/S0001434607090088
4032:
4016:
4003:
3804:can satisfy the equation
3179:is divisible by 4, so is
1241:to these equations gives
943:under the condition that
190:then there is an integer
104:for any integer value of
90:can satisfy the equation
6996:University of St Andrews
5545:. Leipzig: Brandstetter.
5524:Math. Naturwiss. Blätter
5351:, pp. 399, 401–402)
4499:
4351:
4207:(note change in modulus)
4066:Sophie Germain's theorem
3899:Sophie Germain's theorem
3845:, Gambioli, Werebrusow,
3745:
3726:, the original solution
3110:, the original solution
2165:in 1636, 1640 and 1657.
2133:
1125:, where the area equals
994:
961:does divide the product
905:Sophie Germain's theorem
495:where the three numbers
7040:"Fermat's Last Theorem"
6962:"Fermat's Last Theorem"
6930:"Fermat's Last Theorem"
6904:Freeman, Larry (2005).
6062:"Intorno all'equazioni
5470:Proc. R. Soc. Edinburgh
5178:95, July 2011, 269-271.
5159:91, July 2007, 260-262.
4800:Atti Accad. Naz. Lincei
4658:. Gummbinnen: Programm.
4568:Comm. Acad. Sci. Petrop
4292:yields a contradiction
4077:Fermat's little theorem
4068:if the auxiliary prime
3559:we have that 3 divides
2242:published/not published
1943:using Euclid's formula
710:has the same parity as
429:greatest common divisor
269:and for all odd primes
26:, originally stated by
6171:C. R. Acad. Sci. Paris
6137:C. R. Acad. Sci. Paris
6110:C. R. Acad. Sci. Paris
5954:C. R. Acad. Sci. Paris
5444:C. R. Acad. Sci. Paris
5100:. New York: Macmillan.
4597:Novi Acta Acad. Petrop
4374:Victor-Amédée Lebesgue
4329:This section is empty.
4232:However, the equation
3860:Dirichlet's proof for
3776:
3764:
3157:, and one may express
2666:have opposite parity,
2155:
2152:Jakob Emanuel Handmann
1084:infinite descent, see
1009:
985:are pairwise coprime,
681:...−4, −2, 0, 2, 4,...
110:greater than 2. (For
30:in 1637 and proven by
7063:Fermat's Last Theorem
6979:Fermat's last theorem
6703:Mathematische Annalen
6696:Bergmann, G. (1966),
6447:en nombres entiers".
5939:Ribenboim, pp. 57–63.
5725:Ribenboim, pp. 55–57.
5682:Ribenboim, pp. 24–49.
5200:Ribenboim, pp. 11–14.
5125:Reprinted in 1977 in
4863:Reprinted in 1965 in
4532:Ribenboim, pp. 15–24.
4447:each proved the case
4043:after September 1825
3773:Adrien-Marie Legendre
3770:
3758:
3191:is also even. Since
2514:, which implies that
2146:
1047:, since the equation
1007:
626:implies the equation
585:, since the equation
20:Fermat's Last Theorem
6928:Shay, David (2003).
6435:(1974). "L'équation
6410:J. Reine Angew. Math
6079:Ann. Mat. Pura Appl.
5562:Diophantine Analysis
5541:Stockhaus H (1910).
5466:"Mathematical Notes"
5405:Calzolari L (1855).
5175:Mathematical Gazette
5157:Mathematical Gazette
5140:Mathematical Gazette
4916:Kronecker L (1901).
4867:by New York:Chelsea.
4771:Lebesgue VA (1862).
4763:Lebesgue VA (1859).
4547:Mém. Acad. Roy. Sci.
4514:"Fermat's One Proof"
3236:in the equation for
3203:are coprime, so are
2345:with Fermat's note.
1822:are coprime, so are
1697:are both odd, since
146:Factors of exponents
6914:Ribet, Ken (1995).
6801:Elements of Algebra
6323:Amer. Math. Monthly
6177:: 676–679, 743–747.
6021:: 276–279, 348–349.
6015:J. Math. Pures Appl
5981:J. Math. Pures Appl
5893:: 185–195, 305–317.
5857:Moskov. Math. Samml
5804:J. Math. Pures Appl
5770:J. Math. Pures Appl
5450:: 921–924, 961–965.
5063:Amer. Math. Monthly
4967:. Leipzig: Teubner.
4753:J. Math. Pures Appl
4553:, 1666–1699 (1729).
4541:Frénicle de Bessy,
4297:2 ≡ 32 (mod 25) ≡ 7
3983:
2400:Elements of Algebra
2335:a marginal note in
2290:a marginal note in
2232:
1483:In this case, both
1147:Pythagorean theorem
866:are themselves the
797:Prime factorization
685:...−3, −1, 1, 3,...
376:. For example, if
344:Primitive solutions
313:. For example, if
140:Pythagorean triples
6959:Weisstein, Eric W.
6858:. WileyBlackwell.
6776:Notices of the AMS
6743:Brudner, Harvey J.
6720:10.1007/BF01429054
6519:Dover Publications
6370: = 10".
6091:10.1007/BF03198884
5584:Nieuw Archief Wisk
5565:. New York: Wiley.
5305:Mathematical Notes
5096:Hancock H (1931).
5044:Nyt Tidsskrift Mat
5013:Časopis Pěst. Mat.
4946:Nyt Tidsskrift Mat
4243:also implies that
3975:
3777:
3765:
2612:, it follows that
2262:Latin version of
2224:
2156:
1888:Pythagorean triple
1061:can be written as
1010:
909:Since the time of
665:primitive solution
465:may be written as
7018:"The Whole Story"
6924:Taniyama–Shimura.
6865:978-0-471-06261-5
6842:978-0-387-90432-0
6820:978-0-8218-2670-6
6756:978-0-9644785-0-3
6688:978-0-19-513919-8
6669:978-0-88385-451-8
6619:978-0-385-49362-8
6595:978-0-387-98508-4
6559:978-0-387-95002-0
6528:978-0-486-44233-4
6485:978-1-56858-077-7
6169: = 0".
6135: = 0".
5887:Časopis Pěst. Mat
5802: = 0".
5716:Ribenboim, p. 49.
5664: = 0".
5630:Arch. Mat. Naturv
5580:Van der Corput JG
5155: = 4",
5011: = 3".
4963:Sommer J (1907).
4877:Bendz TR (1901).
4349:
4348:
4309:has been proven.
4302:Thus, case A for
4050:
4049:
3466:is even, because
2408:
2407:
2303:1636, 1640, 1657
2169:sent a letter to
1886:form a primitive
1625:that is smaller (
388:are divisible by
73:positive integers
7070:
7043:
7034:
7021:
7012:
6993:
6992:
6991:
6972:
6971:
6953:
6944:
6942:
6941:
6932:. Archived from
6922:
6920:
6909:
6898:
6880:Elkies, Noam D.
6869:
6857:
6846:
6824:
6811:The Fermat Diary
6805:
6791:
6773:
6760:
6738:
6692:
6673:
6660:The Last Problem
6647:
6623:
6606:(October 1998).
6599:
6577:
6563:
6539:
6504:
6489:
6457:
6456:
6429:
6423:
6417:
6402:
6396:
6395:
6366: = 6,
6356:
6347:
6346:
6318:
6312:
6311:
6304:Arch. Math. Phys
6299:
6290:
6289:
6282:Arch. Math. Phys
6277:
6271:
6270:
6246:
6240:
6234:
6219:
6213:
6212:
6185:
6179:
6178:
6151:
6145:
6144:
6117:
6102:
6054:
6048:
6047:
6045:
6044:
6029:
6023:
6022:
5995:
5989:
5988:
5961:
5946:
5940:
5937:
5931:
5930:
5928:
5926:10.5802/aif.1096
5901:
5895:
5894:
5871:
5865:
5864:
5837:
5831:
5830:
5818:
5812:
5811:
5784:
5778:
5777:
5750:
5744:
5743:
5732:
5726:
5723:
5717:
5714:
5708:
5707:
5705:
5704:
5689:
5683:
5680:
5674:
5673:
5649:
5643:
5637:
5598:
5592:
5591:
5576:
5567:
5566:
5553:
5547:
5546:
5538:
5532:
5531:
5519:
5513:
5512:
5492:
5486:
5485:
5458:
5452:
5451:
5428:
5422:
5421:
5402:
5396:
5382:
5367:
5361:
5358:
5352:
5346:
5337:
5336:
5311:(3–4): 352–356.
5300:
5294:
5293:
5291:
5290:
5275:
5269:
5259:
5250:
5245:
5236:
5231:
5216:
5210:
5201:
5198:
5192:
5185:
5179:
5170:descente infinie
5166:
5160:
5149:
5143:
5136:
5130:
5127:Opera matematica
5124:
5117:Gaz. Mat. Ser. A
5108:
5102:
5101:
5093:
5087:
5086:
5058:
5052:
5051:
5027:
5021:
5020:
4997:
4988:
4987:
4975:
4969:
4968:
4960:
4954:
4953:
4929:
4923:
4921:
4913:
4907:
4906:
4894:
4883:
4882:
4874:
4868:
4862:
4845:
4839:
4838:
4831:Ann. Univ. Chile
4814:
4808:
4807:
4783:
4777:
4776:
4768:
4760:
4712:
4706:
4705:
4688:
4682:
4681:
4666:
4660:
4659:
4654:Schopis (1825).
4651:
4645:
4643:
4632:
4623:
4622:
4611:
4605:
4604:
4592:
4581:
4575:
4560:
4554:
4539:
4533:
4530:
4524:
4523:
4521:
4520:
4509:
4495:
4488:
4481:
4474:
4467:
4460:
4453:
4434:
4427:
4420:
4413:
4403:
4396:
4389:
4367:
4357:
4344:
4341:
4331:You can help by
4324:
4317:
4313:Proof for case B
4308:
4298:
4291:
4282:
4265:
4242:
4228:
4206:
4192:
4186:
4176:
4170:
4164:
4158:
4152:
4146:
4137:
4116:
4104:
4092:
4074:
4063:
4053:Proof for case A
3984:
3982:
3974:
3970:
3963:
3956:
3949:
3945:
3941:
3937:
3933:
3929:
3925:
3921:
3917:
3907:
3896:
3882:
3876:
3866:
3821:
3803:
3797:
3791:
3785:
3751:
3742:was impossible.
3741:
3724:infinite descent
3721:
3703:
3688:
3673:
3647:
3637:
3627:
3620:
3614:
3608:
3584:
3582:
3580:
3579:
3576:
3573:
3558:
3552:
3544:
3478:
3471:
3465:
3459:
3450:
3430:
3405:
3388:
3382:
3376:
3365:
3356:
3340:
3327:
3321:
3315:
3304:
3297:
3291:
3285:
3276:
3241:
3235:
3229:
3220:
3214:
3208:
3202:
3196:
3190:
3184:
3178:
3172:
3166:
3156:
3150:
3140:
3130:Proof for case B
3126:was impossible.
3125:
3108:infinite descent
3105:
3087:
3072:
3057:
3043:
3037:
3031:
3025:
3019:
3013:
3001:
2989:
2979:
2945:
2939:
2933:
2927:
2921:
2915:
2910:are coprime, so
2909:
2903:
2895:
2875:
2850:
2834:
2828:
2822:
2808:
2802:
2796:
2783:
2768:
2755:
2749:
2743:
2737:
2727:Proof for case A
2722:
2712:
2705:
2699:
2693:
2687:
2681:
2675:
2665:
2659:
2650:
2611:
2597:
2583:
2577:
2568:
2552:
2534:
2528:
2519:
2513:
2502:
2492:
2486:
2477:
2471:
2465:
2459:
2453:
2439:to the equation
2438:
2421:infinite descent
2418:
2318:
2233:
2231:
2223:
2190:
2179:
2164:
2139:
2130:was impossible.
2129:
2113:
2097:
2082:
2063:
2034:
2024:
2014:
2008:
2002:
1985:
1970:
1955:
1942:
1936:
1927:
1914:
1912:
1911:
1908:
1905:
1885:
1869:
1859:
1849:
1843:
1833:
1827:
1821:
1815:
1806:
1805:
1803:
1802:
1799:
1796:
1777:
1776:
1774:
1773:
1770:
1767:
1746:
1740:
1734:
1728:
1722:
1716:
1710:
1696:
1690:
1681:
1666:
1649:
1640:Proof for case B
1635:
1624:
1605:
1585:
1575:are coprime and
1574:
1568:
1559:
1544:
1529:
1516:
1501:is even. Since
1500:
1494:
1488:
1479:Proof for case A
1474:
1464:
1454:
1448:
1439:
1410:
1400:
1390:
1376:
1358:
1332:
1313:
1287:
1265:
1240:
1222:
1211:
1194:
1189:
1187:
1186:
1183:
1180:
1168:
1162:
1144:
1143:
1141:
1140:
1137:
1134:
1124:
1099:
1082:
1075:
1060:
1046:
1036:
1014:infinite descent
1000:
990:
984:
978:
972:
966:
960:
954:
948:
942:
929:to the equation
928:
891:
881:
871:
865:
859:
853:
847:
838:
825:
812:
808:
792:
786:
780:
774:
768:
762:
746:
733:to the equation
732:
705:
701:
697:
686:
682:
662:
642:
622:
584:
568:
549:
518:
510:
494:
484:
474:
464:
448:
442:
436:
426:
417:
400:
394:
387:
381:
375:
369:
363:
339:
333:
327:
320:
316:
312:
297:
283:
274:
268:
261:
246:
227:
221:
205:
195:
189:
183:
177:
171:
165:
137:
131:
125:
115:
109:
103:
89:
61:infinite descent
58:
51:
45:
28:Pierre de Fermat
22:is a theorem in
7078:
7077:
7073:
7072:
7071:
7069:
7068:
7067:
7048:
7047:
7038:
7029:
7016:
7001:
6989:
6987:
6948:
6939:
6937:
6918:
6876:
6866:
6843:
6821:
6771:
6757:
6689:
6670:
6654:
6652:Further reading
6644:
6620:
6609:Fermat's Enigma
6596:
6560:
6529:
6486:
6465:
6460:
6449:Bull. Sci. Math
6430:
6426:
6403:
6399:
6384:10.2307/3029800
6357:
6350:
6335:10.2307/2972379
6319:
6315:
6300:
6293:
6278:
6274:
6247:
6243:
6220:
6216:
6186:
6182:
6152:
6148:
6118:
6103:
6074: = 0"
6055:
6051:
6042:
6040:
6038: = 7"
6030:
6026:
5996:
5992:
5962:
5947:
5943:
5938:
5934:
5902:
5898:
5872:
5868:
5838:
5834:
5829:: 11–13, 41–42.
5819:
5815:
5785:
5781:
5751:
5747:
5733:
5729:
5724:
5720:
5715:
5711:
5702:
5700:
5690:
5686:
5681:
5677:
5650:
5646:
5599:
5595:
5577:
5570:
5554:
5550:
5539:
5535:
5520:
5516:
5493:
5489:
5459:
5455:
5429:
5425:
5403:
5399:
5368:
5364:
5359:
5355:
5347:
5340:
5301:
5297:
5288:
5286:
5284: = 3"
5276:
5272:
5260:
5253:
5248:Bergmann (1966)
5246:
5239:
5232:
5219:
5211:
5204:
5199:
5195:
5186:
5182:
5167:
5163:
5150:
5146:
5137:
5133:
5109:
5105:
5094:
5090:
5075:10.2307/2974106
5059:
5055:
5028:
5024:
4998:
4991:
4976:
4972:
4961:
4957:
4930:
4926:
4914:
4910:
4895:
4886:
4875:
4871:
4846:
4842:
4815:
4811:
4784:
4780:
4769:
4761:
4713:
4709:
4689:
4685:
4674:Nouv. Ann. Math
4667:
4663:
4652:
4648:
4633:
4626:
4612:
4608:
4593:
4584:
4561:
4557:
4540:
4536:
4531:
4527:
4518:
4516:
4510:
4506:
4502:
4490:
4483:
4476:
4469:
4462:
4455:
4448:
4429:
4428:. Proofs for
4422:
4415:
4408:
4405:
4398:
4391:
4384:
4378:Angelo Genocchi
4362:
4359:
4352:
4345:
4339:
4336:
4315:
4303:
4296:
4287:
4268:
4247:
4233:
4210:
4197:
4188:
4182:
4172:
4166:
4160:
4154:
4148:
4142:
4124:
4107:
4095:
4083:
4069:
4058:
4055:
4030:September 1825
3977:
3965:
3958:
3951:
3947:
3943:
3939:
3935:
3931:
3927:
3923:
3919:
3915:
3902:
3891:
3878:
3872:
3861:
3808:
3799:
3793:
3787:
3780:
3753:
3746:
3727:
3708:
3691:
3676:
3664:
3639:
3629:
3622:
3616:
3610:
3586:
3577:
3574:
3569:
3568:
3566:
3564:
3554:
3550:
3483:
3473:
3467:
3461:
3455:
3433:
3413:
3393:
3384:
3378:
3367:
3361:
3343:
3331:
3323:
3317:
3306:
3299:
3293:
3287:
3281:
3246:
3237:
3231:
3225:
3216:
3210:
3204:
3198:
3192:
3186:
3180:
3174:
3168:
3158:
3152:
3142:
3135:
3132:
3111:
3092:
3075:
3060:
3048:
3039:
3033:
3027:
3021:
3015:
3003:
2991:
2984:
2950:
2941:
2935:
2929:
2923:
2917:
2911:
2905:
2899:
2878:
2858:
2838:
2830:
2824:
2810:
2804:
2798:
2788:
2771:
2759:
2751:
2745:
2739:
2732:
2729:
2714:
2707:
2701:
2695:
2694:is odd. Since
2689:
2683:
2677:
2667:
2661:
2655:
2616:
2599:
2585:
2579:
2573:
2555:
2539:
2530:
2524:
2515:
2504:
2494:
2488:
2482:
2473:
2467:
2461:
2455:
2440:
2424:
2413:
2313:
2226:
2185:
2174:
2159:
2141:
2134:
2115:
2099:
2084:
2068:
2039:
2026:
2016:
2010:
2004:
1990:
1973:
1958:
1947:
1938:
1932:
1909:
1906:
1897:
1896:
1894:
1893:
1871:
1861:
1851:
1845:
1835:
1829:
1823:
1817:
1811:
1800:
1797:
1788:
1787:
1785:
1780:
1771:
1768:
1759:
1758:
1756:
1751:
1742:
1736:
1730:
1724:
1718:
1712:
1698:
1692:
1686:
1669:
1654:
1645:
1642:
1626:
1610:
1590:
1576:
1570:
1564:
1547:
1532:
1521:
1502:
1496:
1490:
1484:
1481:
1466:
1456:
1450:
1444:
1418:
1402:
1392:
1378:
1364:
1341:
1315:
1296:
1267:
1245:
1228:
1213:
1202:
1184:
1181:
1176:
1175:
1173:
1172:
1164:
1150:
1138:
1135:
1130:
1129:
1127:
1126:
1110:
1107:
1087:
1077:
1062:
1048:
1041:
1024:
1002:
995:
986:
980:
974:
968:
962:
956:
950:
944:
930:
914:
907:
901:
883:
873:
867:
861:
855:
849:
843:
830:
821:
810:
806:
799:
788:
782:
776:
775:are both even,
770:
764:
748:
747:. The terms in
734:
718:
703:
699:
695:
684:
680:
677:
648:
630:
589:
570:
554:
523:
516:
496:
486:
476:
466:
450:
444:
438:
432:
422:
405:
396:
389:
383:
377:
371:
365:
349:
346:
340:more apparent.
335:
329:
323:
318:
314:
299:
285:
279:
270:
263:
256:
232:
223:
207:
197:
191:
185:
184:is a factor of
179:
173:
167:
151:
148:
133:
127:
121:
118:linear equation
111:
105:
91:
75:
69:
53:
47:
41:
17:
12:
11:
5:
7076:
7066:
7065:
7060:
7058:Article proofs
7046:
7045:
7036:
7027:
7014:
6999:
6973:
6954:
6946:
6925:
6911:
6901:
6875:
6874:External links
6872:
6871:
6870:
6864:
6847:
6841:
6825:
6819:
6806:
6792:
6782:(7): 743–746.
6761:
6755:
6739:
6693:
6687:
6674:
6668:
6653:
6650:
6649:
6648:
6642:
6624:
6618:
6600:
6594:
6578:
6564:
6558:
6548:(2008-05-23).
6542:
6541:
6540:
6527:
6490:
6484:
6472:(1996-09-30).
6464:
6461:
6459:
6458:
6424:
6397:
6378:(5): 279–281.
6348:
6313:
6291:
6272:
6241:
6214:
6180:
6146:
6049:
6024:
5990:
5941:
5932:
5896:
5866:
5832:
5813:
5779:
5745:
5727:
5718:
5709:
5684:
5675:
5644:
5593:
5568:
5548:
5533:
5514:
5487:
5453:
5423:
5397:
5362:
5353:
5338:
5295:
5270:
5251:
5237:
5217:
5215:, p. 546)
5202:
5193:
5180:
5161:
5144:
5131:
5103:
5088:
5069:(7): 213–221.
5053:
5022:
5007: = 4 and
4989:
4970:
4955:
4924:
4908:
4884:
4869:
4840:
4809:
4778:
4731: = 2
4723: ± 2
4707:
4683:
4661:
4646:
4624:
4606:
4582:
4555:
4534:
4525:
4503:
4501:
4498:
4404:
4382:
4368:was proven by
4358:
4350:
4347:
4346:
4327:
4325:
4314:
4311:
4300:
4299:
4284:
4283:
4266:
4230:
4229:
4208:
4139:
4138:
4120:and therefore
4118:
4117:
4105:
4093:
4054:
4051:
4048:
4047:
4044:
4040:
4039:
4034:
4031:
4027:
4026:
4021:
4018:
4015:
4011:
4010:
4005:
4002:
3998:
3997:
3994:
3991:
3988:
3973:
3972:
3909:
3869:Sophie Germain
3851:van der Corput
3823:
3822:
3771:Caricature of
3752:
3744:
3705:
3704:
3689:
3674:
3547:
3546:
3452:
3451:
3431:
3407:
3406:
3358:
3357:
3341:
3278:
3277:
3131:
3128:
3089:
3088:
3073:
3058:
2981:
2980:
2946:is odd. Since
2897:
2896:
2876:
2852:
2851:
2797:is odd, so is
2785:
2784:
2769:
2728:
2725:
2652:
2651:
2570:
2569:
2553:
2410:
2409:
2406:
2405:
2402:
2396:
2393:
2390:
2386:
2385:
2382:
2379:
2376:
2373:
2369:
2368:
2363:
2357:
2354:
2351:
2350:4 August 1753
2347:
2346:
2339:
2333:
2330:
2327:
2323:
2322:
2319:
2310:
2307:
2304:
2300:
2299:
2294:
2288:
2285:
2282:
2278:
2277:
2272:
2260:
2257:
2254:
2250:
2249:
2246:
2243:
2240:
2237:
2219:, and Duarte.
2213:van der Corput
2148:Leonhard Euler
2140:
2132:
2065:
2064:
1987:
1986:
1971:
1956:
1929:
1928:
1808:
1807:
1778:
1683:
1682:
1667:
1641:
1638:
1607:
1606:
1561:
1560:
1545:
1530:
1480:
1477:
1441:
1440:
1361:
1360:
1335:
1334:
1290:
1289:
1225:
1224:
1196:
1195:
1106:
1103:
1038:
1037:
1001:
993:
911:Sophie Germain
903:Main article:
900:
897:
840:
839:
798:
795:
676:
673:
645:
644:
624:
623:
551:
550:
427:represent the
419:
418:
345:
342:
255:solutions for
249:
248:
147:
144:
68:
65:
15:
9:
6:
4:
3:
2:
7075:
7064:
7061:
7059:
7056:
7055:
7053:
7041:
7037:
7032:
7028:
7025:
7019:
7015:
7010:
7009:
7004:
7000:
6997:
6986:on 2013-01-16
6985:
6981:
6980:
6974:
6969:
6968:
6963:
6960:
6955:
6951:
6947:
6936:on 2012-02-27
6935:
6931:
6926:
6917:
6912:
6907:
6902:
6900:
6897:
6894:
6893:
6890: =
6889:
6886: +
6885:
6878:
6877:
6867:
6861:
6856:
6855:
6848:
6844:
6838:
6834:
6830:
6826:
6822:
6816:
6812:
6807:
6803:
6802:
6797:
6793:
6789:
6785:
6781:
6777:
6770:
6767:(July 1995).
6766:
6762:
6758:
6752:
6748:
6744:
6740:
6737:
6733:
6729:
6725:
6721:
6717:
6713:
6709:
6705:
6704:
6699:
6694:
6690:
6684:
6680:
6675:
6671:
6665:
6661:
6656:
6655:
6645:
6643:0-262-69060-8
6639:
6636:. MIT Press.
6635:
6634:
6629:
6625:
6621:
6615:
6611:
6610:
6605:
6601:
6597:
6591:
6587:
6583:
6579:
6575:
6574:
6569:
6565:
6561:
6555:
6551:
6547:
6543:
6538:
6534:
6530:
6524:
6520:
6516:
6515:
6510:
6506:
6505:
6502:
6500:
6495:
6491:
6487:
6481:
6477:
6476:
6471:
6467:
6466:
6454:
6450:
6446:
6443: =
6442:
6439: +
6438:
6434:
6428:
6421:
6418:Reprinted in
6415:
6411:
6407:
6406:Dirichlet PGL
6401:
6393:
6389:
6385:
6381:
6377:
6373:
6369:
6365:
6361:
6355:
6353:
6344:
6340:
6336:
6332:
6328:
6324:
6317:
6309:
6305:
6298:
6296:
6287:
6283:
6276:
6268:
6264:
6260:
6257: =
6256:
6253: +
6252:
6245:
6238:
6235:Reprinted in
6232:
6228:
6224:
6218:
6210:
6206:
6202:
6199: =
6198:
6195: +
6194:
6190:
6184:
6176:
6172:
6168:
6165: +
6164:
6161: +
6160:
6156:
6150:
6142:
6138:
6134:
6131: +
6130:
6127: +
6126:
6122:
6115:
6111:
6107:
6100:
6096:
6092:
6088:
6084:
6081:
6080:
6075:
6073:
6070: +
6069:
6066: +
6065:
6059:
6053:
6039:
6037:
6028:
6020:
6016:
6012:
6009: +
6008:
6005: +
6004:
6000:
5994:
5986:
5982:
5978:
5975: =
5974:
5971: +
5970:
5966:
5959:
5955:
5951:
5945:
5936:
5927:
5922:
5918:
5914:
5910:
5906:
5900:
5892:
5888:
5884:
5883:(in Bohemian)
5880:
5876:
5870:
5862:
5858:
5854:
5851:
5847:
5843:
5836:
5828:
5824:
5817:
5809:
5805:
5801:
5798: +
5797:
5794: +
5793:
5789:
5783:
5775:
5771:
5767:
5764: =
5763:
5760: +
5759:
5755:
5749:
5741:
5737:
5731:
5722:
5713:
5699:
5697:
5688:
5679:
5671:
5667:
5663:
5660: +
5659:
5656: +
5655:
5648:
5641:
5638:Reprinted in
5635:
5631:
5627:
5623:
5619:
5615:
5612: =
5611:
5608: +
5607:
5603:
5597:
5589:
5585:
5581:
5575:
5573:
5564:
5563:
5558:
5557:Carmichael RD
5552:
5544:
5537:
5529:
5525:
5518:
5510:
5506:
5503: =
5502:
5499: +
5498:
5491:
5483:
5479:
5475:
5471:
5467:
5463:
5457:
5449:
5445:
5441:
5438: ±
5437:
5433:
5427:
5419:
5416:
5415: =
5412:
5411: +
5408:
5401:
5394:
5390:
5389:Sphinx-Oedipe
5386:
5380:
5376:
5372:
5366:
5357:
5350:
5345:
5343:
5334:
5330:
5326:
5322:
5318:
5314:
5310:
5306:
5299:
5285:
5283:
5274:
5267:
5263:
5258:
5256:
5249:
5244:
5242:
5235:
5230:
5228:
5226:
5224:
5222:
5214:
5213:Dickson (2005
5209:
5207:
5197:
5190:
5184:
5177:
5176:
5171:
5165:
5158:
5154:
5148:
5141:
5135:
5128:
5122:
5118:
5114:
5107:
5099:
5092:
5084:
5080:
5076:
5072:
5068:
5064:
5057:
5049:
5045:
5041:
5038: =
5037:
5034: +
5033:
5026:
5018:
5014:
5010:
5006:
5002:
4996:
4994:
4985:
4981:
4974:
4966:
4959:
4951:
4947:
4943:
4940: =
4939:
4936: −
4935:
4928:
4919:
4912:
4904:
4900:
4893:
4891:
4889:
4880:
4873:
4866:
4860:
4856:
4855:
4850:
4844:
4836:
4832:
4828:
4825: =
4824:
4821: +
4820:
4813:
4805:
4801:
4797:
4794: =
4793:
4790: +
4789:
4782:
4774:
4766:
4758:
4754:
4750:
4747: ±
4746:
4743: =
4742:
4738:
4735: −
4734:
4730:
4726:
4722:
4719: =
4718:
4711:
4703:
4699:
4698:
4693:
4687:
4679:
4675:
4671:
4665:
4657:
4650:
4641:
4637:
4631:
4629:
4620:
4616:
4610:
4602:
4598:
4591:
4589:
4587:
4579:
4576:. Reprinted
4573:
4569:
4565:
4559:
4552:
4548:
4544:
4538:
4529:
4515:
4508:
4504:
4497:
4493:
4486:
4479:
4472:
4465:
4458:
4451:
4446:
4442:
4438:
4432:
4425:
4418:
4411:
4401:
4394:
4387:
4381:
4379:
4375:
4371:
4365:
4355:
4343:
4334:
4330:
4326:
4323:
4319:
4318:
4310:
4306:
4295:
4294:
4293:
4290:
4280:
4276:
4272:
4267:
4263:
4259:
4255:
4251:
4246:
4245:
4244:
4240:
4236:
4226:
4222:
4218:
4214:
4209:
4204:
4200:
4196:
4195:
4194:
4191:
4185:
4179:
4175:
4169:
4163:
4157:
4151:
4145:
4135:
4131:
4127:
4123:
4122:
4121:
4114:
4110:
4106:
4102:
4098:
4094:
4090:
4086:
4082:
4081:
4080:
4078:
4072:
4067:
4061:
4045:
4042:
4041:
4038:
4035:
4029:
4028:
4025:
4022:
4019:
4013:
4012:
4009:
4006:
4000:
3999:
3995:
3993:case II(i/ii)
3992:
3989:
3986:
3985:
3980:
3968:
3961:
3954:
3913:
3910:
3905:
3900:
3894:
3889:
3886:
3885:
3884:
3881:
3875:
3870:
3864:
3858:
3856:
3852:
3848:
3844:
3840:
3836:
3832:
3828:
3819:
3815:
3811:
3807:
3806:
3805:
3802:
3796:
3790:
3783:
3774:
3769:
3762:
3757:
3749:
3743:
3739:
3735:
3731:
3725:
3719:
3715:
3711:
3702:
3698:
3694:
3690:
3687:
3683:
3679:
3675:
3672:
3668:
3663:
3662:
3661:
3659:
3655:
3651:
3646:
3642:
3636:
3632:
3626:
3619:
3613:
3606:
3602:
3598:
3594:
3590:
3572:
3562:
3557:
3542:
3538:
3534:
3530:
3526:
3522:
3518:
3514:
3510:
3506:
3502:
3498:
3494:
3490:
3486:
3482:
3481:
3480:
3479:then becomes
3477:
3470:
3464:
3458:
3448:
3444:
3440:
3436:
3432:
3428:
3424:
3420:
3416:
3412:
3411:
3410:
3404:
3400:
3396:
3392:
3391:
3390:
3387:
3381:
3375:
3371:
3364:
3355:
3351:
3347:
3342:
3339:
3335:
3330:
3329:
3328:
3326:
3320:
3314:
3310:
3303:
3296:
3290:
3284:
3274:
3270:
3266:
3262:
3258:
3254:
3250:
3245:
3244:
3243:
3240:
3234:
3228:
3224:Substituting
3222:
3219:
3213:
3207:
3201:
3195:
3189:
3183:
3177:
3171:
3165:
3161:
3155:
3149:
3145:
3139:
3127:
3123:
3119:
3115:
3109:
3103:
3099:
3095:
3086:
3082:
3078:
3074:
3071:
3067:
3063:
3059:
3056:
3052:
3047:
3046:
3045:
3042:
3036:
3030:
3024:
3018:
3011:
3007:
2999:
2995:
2988:
2977:
2973:
2969:
2965:
2961:
2957:
2953:
2949:
2948:
2947:
2944:
2938:
2932:
2926:
2920:
2914:
2908:
2902:
2893:
2889:
2885:
2881:
2877:
2873:
2869:
2865:
2861:
2857:
2856:
2855:
2849:
2845:
2841:
2837:
2836:
2835:
2833:
2827:
2821:
2817:
2813:
2807:
2801:
2795:
2791:
2782:
2778:
2774:
2770:
2767:
2763:
2758:
2757:
2756:
2754:
2748:
2742:
2736:
2724:
2721:
2717:
2711:
2704:
2698:
2692:
2686:
2680:
2674:
2670:
2664:
2658:
2648:
2644:
2640:
2636:
2632:
2628:
2624:
2620:
2615:
2614:
2613:
2610:
2606:
2602:
2596:
2592:
2588:
2582:
2576:
2567:
2563:
2559:
2554:
2551:
2547:
2543:
2538:
2537:
2536:
2533:
2527:
2521:
2518:
2512:
2508:
2501:
2497:
2491:
2485:
2479:
2476:
2470:
2464:
2458:
2451:
2447:
2443:
2436:
2432:
2428:
2422:
2416:
2403:
2401:
2397:
2394:
2391:
2388:
2387:
2383:
2380:
2378:not published
2377:
2374:
2371:
2370:
2367:
2364:
2362:
2358:
2355:
2352:
2349:
2348:
2344:
2340:
2338:
2334:
2331:
2328:
2325:
2324:
2320:
2316:
2311:
2308:
2305:
2302:
2301:
2298:
2295:
2293:
2289:
2287:not published
2286:
2283:
2280:
2279:
2276:
2273:
2271:
2270:
2265:
2261:
2258:
2255:
2252:
2251:
2247:
2244:
2241:
2238:
2235:
2234:
2229:
2222:
2221:
2220:
2218:
2214:
2210:
2207:, Stockhaus,
2206:
2202:
2198:
2195:, Calzolari,
2194:
2188:
2183:
2177:
2172:
2168:
2162:
2153:
2149:
2145:
2137:
2131:
2127:
2123:
2119:
2111:
2107:
2103:
2096:
2092:
2088:
2080:
2076:
2072:
2067:The solution
2062:
2058:
2054:
2050:
2046:
2042:
2038:
2037:
2036:
2033:
2029:
2023:
2019:
2013:
2007:
2001:
1997:
1993:
1984:
1980:
1976:
1972:
1969:
1965:
1961:
1957:
1954:
1950:
1946:
1945:
1944:
1941:
1935:
1926:
1922:
1918:
1904:
1900:
1892:
1891:
1890:
1889:
1883:
1879:
1875:
1868:
1864:
1858:
1854:
1848:
1842:
1838:
1832:
1826:
1820:
1814:
1795:
1791:
1783:
1779:
1766:
1762:
1754:
1750:
1749:
1748:
1745:
1739:
1733:
1727:
1721:
1715:
1709:
1705:
1701:
1695:
1689:
1680:
1676:
1672:
1668:
1665:
1661:
1657:
1653:
1652:
1651:
1648:
1637:
1634:
1630:
1622:
1618:
1614:
1604:
1600:
1596:
1593:
1589:
1588:
1587:
1583:
1579:
1573:
1567:
1558:
1554:
1550:
1546:
1543:
1539:
1535:
1531:
1528:
1524:
1520:
1519:
1518:
1514:
1510:
1506:
1499:
1493:
1487:
1476:
1473:
1469:
1463:
1459:
1453:
1447:
1438:
1434:
1430:
1426:
1422:
1417:
1416:
1415:
1412:
1409:
1405:
1399:
1395:
1389:
1385:
1381:
1375:
1371:
1367:
1357:
1353:
1349:
1345:
1340:
1339:
1338:
1331:
1327:
1323:
1319:
1312:
1308:
1304:
1300:
1295:
1294:
1293:
1286:
1282:
1278:
1274:
1270:
1264:
1260:
1256:
1252:
1248:
1244:
1243:
1242:
1239:
1235:
1231:
1221:
1217:
1210:
1206:
1201:
1200:
1199:
1193:
1179:
1171:
1170:
1169:
1167:
1161:
1157:
1153:
1148:
1133:
1122:
1118:
1114:
1102:
1100:
1098:
1094:
1090:
1080:
1073:
1069:
1065:
1059:
1055:
1051:
1044:
1035:
1031:
1027:
1023:
1022:
1021:
1019:
1015:
1006:
998:
992:
989:
983:
977:
971:
965:
959:
953:
947:
941:
937:
933:
926:
922:
918:
912:
906:
896:
893:
890:
886:
880:
876:
870:
864:
858:
852:
846:
837:
833:
829:
828:
827:
824:
818:
816:
815:number theory
804:
794:
791:
785:
779:
773:
767:
760:
756:
752:
745:
741:
737:
730:
726:
722:
715:
713:
709:
692:
690:
672:
670:
666:
660:
656:
652:
641:
637:
633:
629:
628:
627:
621:
618:
614:
611:
607:
604:
600:
596:
592:
588:
587:
586:
582:
578:
574:
566:
562:
558:
547:
543:
539:
535:
531:
527:
522:
521:
520:
514:
511:are pairwise
508:
504:
500:
493:
489:
483:
479:
473:
469:
462:
458:
454:
447:
441:
435:
430:
425:
416:
412:
408:
404:
403:
402:
399:
392:
386:
380:
374:
368:
361:
357:
353:
341:
338:
332:
326:
310:
306:
302:
296:
292:
288:
282:
276:
273:
266:
259:
254:
244:
240:
236:
231:
230:
229:
226:
219:
215:
211:
204:
200:
194:
188:
182:
176:
170:
163:
159:
155:
143:
141:
136:
130:
124:
119:
114:
108:
102:
98:
94:
87:
83:
79:
74:
64:
62:
56:
50:
44:
40:
37:
33:
29:
25:
24:number theory
21:
7023:
7006:
6988:, retrieved
6984:the original
6978:
6965:
6938:. Retrieved
6934:the original
6899:
6895:
6891:
6887:
6883:
6853:
6832:
6810:
6800:
6779:
6775:
6749:. WLC, Inc.
6746:
6707:
6701:
6678:
6659:
6632:
6607:
6585:
6572:
6549:
6517:, New York:
6513:
6497:
6474:
6452:
6448:
6444:
6440:
6436:
6427:
6419:
6413:
6409:
6400:
6375:
6371:
6367:
6363:
6326:
6322:
6316:
6307:
6303:
6285:
6281:
6275:
6266:
6262:
6258:
6254:
6250:
6244:
6236:
6230:
6226:
6217:
6208:
6207:. Série II.
6204:
6200:
6196:
6192:
6183:
6174:
6170:
6166:
6162:
6158:
6149:
6140:
6136:
6132:
6128:
6124:
6113:
6109:
6082:
6077:
6071:
6067:
6063:
6052:
6041:. Retrieved
6035:
6027:
6018:
6014:
6010:
6006:
6002:
5993:
5984:
5980:
5976:
5972:
5968:
5957:
5953:
5944:
5935:
5919:(3): 19–37.
5916:
5912:
5899:
5890:
5886:
5882:
5878:
5869:
5860:
5856:
5853:(in Russian)
5852:
5849:
5845:
5841:
5835:
5826:
5822:
5816:
5807:
5803:
5799:
5795:
5791:
5782:
5773:
5769:
5765:
5761:
5757:
5748:
5739:
5730:
5721:
5712:
5701:. Retrieved
5695:
5687:
5678:
5669:
5665:
5661:
5657:
5653:
5647:
5639:
5633:
5629:
5625:
5621:
5617:
5613:
5609:
5605:
5596:
5587:
5583:
5561:
5551:
5542:
5536:
5527:
5523:
5517:
5508:
5504:
5500:
5496:
5490:
5473:
5469:
5456:
5447:
5443:
5439:
5435:
5426:
5418:
5414:
5410:
5406:
5400:
5392:
5388:
5384:
5378:
5374:
5365:
5356:
5308:
5304:
5298:
5287:. Retrieved
5281:
5273:
5265:
5196:
5188:
5183:
5173:
5169:
5164:
5156:
5152:
5147:
5139:
5134:
5126:
5120:
5116:
5112:
5106:
5097:
5091:
5066:
5062:
5056:
5047:
5043:
5039:
5035:
5031:
5025:
5016:
5015:(in Czech).
5012:
5008:
5004:
4983:
4979:
4973:
4964:
4958:
4949:
4945:
4941:
4937:
4933:
4927:
4917:
4911:
4902:
4898:
4878:
4872:
4864:
4858:
4852:
4843:
4834:
4830:
4826:
4822:
4818:
4812:
4803:
4799:
4795:
4791:
4787:
4781:
4772:
4764:
4756:
4752:
4748:
4744:
4740:
4736:
4732:
4728:
4724:
4720:
4716:
4710:
4696:
4692:Bertrand JLF
4686:
4677:
4673:
4664:
4655:
4649:
4639:
4618:
4609:
4600:
4596:
4577:
4571:
4567:
4558:
4550:
4546:
4542:
4537:
4528:
4517:. Retrieved
4507:
4491:
4484:
4477:
4470:
4463:
4456:
4449:
4430:
4423:
4416:
4409:
4406:
4399:
4392:
4385:
4370:Gabriel Lamé
4363:
4360:
4353:
4340:January 2011
4337:
4333:adding to it
4328:
4304:
4301:
4288:
4285:
4278:
4274:
4270:
4261:
4257:
4253:
4249:
4238:
4234:
4231:
4224:
4220:
4216:
4212:
4202:
4198:
4189:
4183:
4180:
4173:
4167:
4161:
4155:
4149:
4143:
4140:
4133:
4129:
4125:
4119:
4112:
4108:
4100:
4096:
4088:
4084:
4070:
4059:
4056:
4033:case II(ii)
3978:
3966:
3959:
3952:
3903:
3892:
3879:
3873:
3862:
3859:
3824:
3817:
3813:
3809:
3800:
3794:
3788:
3781:
3778:
3759:Portrait of
3747:
3737:
3733:
3729:
3717:
3713:
3709:
3706:
3700:
3696:
3692:
3685:
3681:
3677:
3670:
3666:
3657:
3653:
3649:
3644:
3640:
3634:
3630:
3624:
3617:
3611:
3604:
3600:
3596:
3592:
3588:
3570:
3560:
3555:
3548:
3540:
3536:
3532:
3528:
3524:
3520:
3516:
3512:
3508:
3504:
3500:
3496:
3492:
3488:
3484:
3475:
3468:
3462:
3456:
3453:
3446:
3442:
3438:
3434:
3426:
3422:
3418:
3414:
3408:
3402:
3398:
3394:
3385:
3379:
3373:
3369:
3362:
3359:
3353:
3349:
3345:
3337:
3333:
3324:
3318:
3312:
3308:
3301:
3294:
3288:
3282:
3279:
3272:
3268:
3264:
3260:
3256:
3252:
3248:
3238:
3232:
3226:
3223:
3217:
3211:
3205:
3199:
3193:
3187:
3181:
3175:
3169:
3163:
3159:
3153:
3147:
3143:
3137:
3133:
3121:
3117:
3113:
3101:
3097:
3093:
3090:
3084:
3080:
3076:
3069:
3065:
3061:
3054:
3050:
3040:
3034:
3028:
3022:
3016:
3009:
3005:
2997:
2993:
2986:
2983:The factors
2982:
2975:
2971:
2967:
2963:
2959:
2955:
2951:
2942:
2940:is even and
2936:
2930:
2928:is even and
2924:
2918:
2912:
2906:
2900:
2898:
2891:
2887:
2883:
2879:
2871:
2867:
2863:
2859:
2853:
2847:
2843:
2839:
2831:
2825:
2819:
2815:
2811:
2805:
2799:
2793:
2789:
2786:
2780:
2776:
2772:
2765:
2761:
2752:
2746:
2740:
2734:
2730:
2719:
2715:
2709:
2702:
2696:
2690:
2688:is even and
2684:
2678:
2672:
2668:
2662:
2656:
2653:
2646:
2642:
2638:
2634:
2630:
2626:
2622:
2618:
2608:
2604:
2600:
2594:
2590:
2586:
2580:
2574:
2571:
2565:
2561:
2557:
2549:
2545:
2541:
2531:
2525:
2522:
2516:
2510:
2506:
2499:
2495:
2489:
2483:
2480:
2474:
2468:
2462:
2456:
2449:
2445:
2441:
2434:
2430:
2426:
2414:
2411:
2399:
2342:
2336:
2314:
2291:
2281:around 1630
2267:
2239:result/proof
2227:
2186:
2175:
2160:
2157:
2135:
2125:
2121:
2117:
2109:
2105:
2101:
2094:
2090:
2086:
2078:
2074:
2070:
2066:
2060:
2056:
2052:
2048:
2044:
2040:
2031:
2027:
2021:
2017:
2011:
2005:
2003:, and since
1999:
1995:
1991:
1988:
1982:
1978:
1974:
1967:
1963:
1959:
1952:
1948:
1939:
1933:
1930:
1924:
1920:
1916:
1902:
1898:
1881:
1877:
1873:
1866:
1862:
1856:
1852:
1846:
1840:
1836:
1830:
1824:
1818:
1812:
1809:
1793:
1789:
1781:
1764:
1760:
1752:
1743:
1737:
1731:
1725:
1719:
1713:
1707:
1703:
1699:
1693:
1687:
1685:The numbers
1684:
1678:
1674:
1670:
1663:
1659:
1655:
1646:
1643:
1632:
1628:
1620:
1616:
1612:
1608:
1602:
1598:
1594:
1591:
1581:
1577:
1571:
1565:
1562:
1556:
1552:
1548:
1541:
1537:
1533:
1526:
1522:
1512:
1508:
1504:
1497:
1495:are odd and
1491:
1485:
1482:
1471:
1467:
1461:
1457:
1451:
1445:
1442:
1436:
1432:
1428:
1424:
1420:
1413:
1407:
1403:
1397:
1393:
1387:
1383:
1379:
1373:
1369:
1365:
1362:
1355:
1351:
1347:
1343:
1336:
1329:
1325:
1321:
1317:
1310:
1306:
1302:
1298:
1291:
1284:
1280:
1276:
1272:
1268:
1262:
1258:
1254:
1250:
1246:
1237:
1233:
1229:
1226:
1219:
1215:
1208:
1204:
1197:
1191:
1177:
1165:
1159:
1155:
1151:
1145:and, by the
1131:
1120:
1116:
1112:
1108:
1096:
1092:
1088:
1078:
1071:
1067:
1063:
1057:
1053:
1049:
1042:
1039:
1033:
1029:
1025:
1011:
996:
987:
981:
975:
969:
963:
957:
951:
945:
939:
935:
931:
924:
920:
916:
908:
894:
888:
884:
878:
874:
868:
862:
856:
850:
844:
841:
835:
831:
822:
819:
800:
789:
783:
777:
771:
765:
758:
754:
750:
743:
739:
735:
728:
724:
720:
716:
711:
707:
693:
678:
675:Even and odd
668:
664:
663:is called a
658:
654:
650:
646:
639:
635:
631:
625:
619:
616:
612:
609:
605:
602:
598:
594:
590:
580:
576:
572:
564:
560:
556:
552:
545:
541:
537:
533:
529:
525:
506:
502:
498:
491:
487:
481:
477:
471:
467:
460:
456:
452:
445:
439:
433:
423:
420:
414:
410:
406:
397:
390:
384:
378:
372:
366:
359:
355:
351:
347:
336:
330:
324:
308:
304:
300:
294:
290:
286:
280:
277:
271:
264:
257:
252:
250:
242:
238:
234:
224:
217:
213:
209:
202:
198:
192:
186:
180:
174:
168:
166:for a given
161:
157:
153:
149:
134:
128:
122:
112:
106:
100:
96:
92:
85:
81:
77:
70:
54:
48:
42:
32:Andrew Wiles
18:
7003:"The Proof"
6829:Ribenboim P
6714:: 159–175,
6582:Ribenboim P
6546:Edwards, HM
6509:Dickson, LE
6470:Aczel, Amir
6451:. Série 2.
6433:Terjanian G
6329:: 238–239.
6085:: 287–288.
6032:Freeman L.
5999:Lebesgue VA
5905:Terjanian G
5823:Il Pitagora
5754:Lebesgue VA
5692:Freeman L.
5371:Legendre AM
5349:Euler (1822
5278:Freeman L.
4980:Period. Mat
4899:Period. Mat
4636:Legendre AM
4578:Opera omnia
4512:Freeman L.
4136:≡ 0 (mod 5)
4057:Case A for
4020:case II(i)
3460:is odd and
2353:only result
2343:Arithmetica
2337:Arithmetica
2329:only result
2312:letters of
2306:only result
2292:Arithmetica
2284:only result
2269:Arithmetica
2184:proof for
150:A solution
7052:Categories
7024:Prometheus
6990:2009-06-02
6940:2004-08-05
6765:Faltings G
6736:0138.25101
6568:Mordell LJ
6494:Dickson LE
6463:References
6416:: 390–393.
6310:: 143–146.
6288:: 368–369.
6211:: 156–168.
6143:: 910–913.
6121:Genocchi A
6116:: 433–436.
6106:Genocchi A
6058:Genocchi A
6043:2009-05-23
5987:: 195–211.
5863:: 466–473.
5810:: 137–171.
5703:2009-05-23
5672:: 971–979.
5530:: 179–180.
5511:: 112–120.
5420:. Ferrara.
5289:2009-05-23
5123:: 334–335.
4986:: 104–110.
4905:: 145–192.
4861:: 175–546.
4837:: 307–320.
4704:–230, 395.
4603:: 245–253.
4574:: 125–146.
4519:2009-05-23
4046:Dirichlet
4014:July 1825
2359:letter to
2264:Diophantus
2209:Carmichael
817:is built.
704:3 + 8 = 11
696:4 + 6 = 10
319:(3, 5, −8)
196:such that
6967:MathWorld
6796:Euler, L.
6788:0002-9920
6728:119984911
6511:(2005) ,
6372:Math. Mag
6360:Breusch R
6189:Maillet E
6099:124916552
5875:Rychlik K
5395:, 97–128.
5333:121798358
5001:Rychlík K
4849:Hilbert D
4670:Terquem O
4445:Terjanian
4441:Dirichlet
4361:The case
4024:Dirichlet
3990:case I/II
3855:Terjanian
3827:Dirichlet
3523:) = 3 × 2
3185:; hence,
3173:. Since
2682:is even,
2395:published
2356:published
2332:published
2309:published
2259:published
1870:. Since
1586:. Thus,
967:. Since
899:Two cases
811:7 × 3 × 2
807:2 × 3 × 7
700:3 + 5 = 8
315:(3, 5, 8)
6831:(1979).
6798:(1822),
6745:(1994).
6712:Springer
6630:(1978).
6584:(2000).
6570:(1921).
6496:(1919).
6455:: 91–95.
6269:: 63–80.
6060:(1864).
5960:: 45–46.
5907:(1987).
5776:: 49–70.
5736:Gauss CF
5590:: 45–75.
5559:(1915).
5464:(1872).
5050:: 33–38.
5019:: 65–86.
4952:: 35–36.
4806:: 34–70.
4759:: 73–86.
4694:(1851).
4680:: 70–87.
4638:(1830).
4617:(1811).
4615:Barlow P
4281:(mod 25)
4227:(mod 25)
4205:(mod 25)
4037:Legendre
3839:Lebesgue
3831:Legendre
3609:. Since
3553:divides
3280:Because
2854:so that
2361:Goldbach
2193:Legendre
2171:Goldbach
540:) = GCD(
532:) = GCD(
449:. Then
206:. Then
39:exponent
6628:Stark H
6604:Singh S
6537:0245500
6392:3029800
6343:2972379
6155:Pépin T
5476:: 144.
5462:Tait PG
5381:: 1–60.
5325:2364600
5264:(1770)
5262:Euler L
5083:2974106
4564:Euler L
4264:(mod 5)
4241:(mod 5)
4115:(mod 5)
4103:(mod 5)
4091:(mod 5)
4017:case II
4008:Germain
4004:case I
3912:Case II
3847:Rychlik
3581:
3567:
3298:, then
3242:yields
2503:, then
2321:Fermat
2205:Rychlik
2182:Euler's
2085:0 <
1913:
1895:
1804:
1786:
1775:
1757:
1627:0 <
1227:Adding
1188:
1174:
1142:
1128:
842:and if
513:coprime
395:, then
132:. For
36:integer
6862:
6839:
6817:
6786:
6753:
6734:
6726:
6685:
6666:
6640:
6616:
6592:
6556:
6535:
6525:
6482:
6390:
6341:
6223:Thue A
6097:
5965:Lamé G
5950:Lamé G
5788:Lamé G
5602:Thue A
5432:Lamé G
5331:
5323:
5189:Œuvres
5081:
4421:, and
4397:, and
4153:, and
3888:Case I
3853:, and
3656:, and
3638:, and
3549:Since
3503:) = 54
3454:Thus,
3263:) = 18
3002:, and
2787:Since
2654:Since
2523:Since
2481:Since
2466:, and
2404:Euler
2384:Euler
2297:Fermat
2275:Bachet
1989:Since
1810:Since
1584:> 0
1563:where
1443:Since
979:, and
689:parity
485:, and
443:, and
260:> 2
6919:(PDF)
6772:(PDF)
6724:S2CID
6710:(2),
6420:Werke
6388:JSTOR
6339:JSTOR
6095:S2CID
5636:(15).
5329:S2CID
5115:=4".
5079:JSTOR
4500:Notes
4001:1823
3996:name
3835:Gauss
3563:, so
3020:: if
2934:odd,
2637:) = 2
2629:) + (
2392:proof
2389:1770
2375:proof
2372:1760
2366:Euler
2326:1670
2253:1621
2248:name
2167:Euler
2093:<
2089:<
1631:<
1580:>
548:) = 1
241:) = (
237:) + (
178:: if
6860:ISBN
6837:ISBN
6815:ISBN
6784:ISSN
6751:ISBN
6683:ISBN
6664:ISBN
6638:ISBN
6614:ISBN
6590:ISBN
6554:ISBN
6523:ISBN
6480:ISBN
5881:= 5
5698:= 5"
4487:= 14
4459:= 10
4452:= 14
4443:and
4437:Thue
4426:= 14
4419:= 10
4402:= 14
4395:= 10
4277:≡ 32
4187:and
4171:and
4073:= 11
3987:date
3906:= 11
3890:for
3843:Lamé
3829:and
3798:and
3615:and
3491:= 54
3487:= 18
3383:and
3322:and
3305:and
3286:and
3209:and
3197:and
3141:and
3038:and
2916:and
2904:and
2829:and
2750:and
2713:and
2700:and
2660:and
2598:and
2578:and
2529:and
2487:and
2256:none
2245:work
2236:date
2217:Thue
2201:Tait
2197:Lamé
2025:and
2009:and
1937:and
1860:and
1828:and
1816:and
1741:and
1729:and
1717:and
1691:and
1569:and
1489:and
1465:and
1449:and
1435:) =
1401:and
1354:− 16
1350:) =
1324:) =
1314:and
1305:) =
1266:and
1218:= −4
1212:and
882:and
860:and
848:and
801:The
787:and
769:and
698:and
524:GCD(
393:= 13
382:and
334:and
126:and
7008:PBS
6892:z'"
6732:Zbl
6716:doi
6708:164
6380:doi
6331:doi
6203:".
6087:doi
5921:doi
5885:".
5855:".
5768:".
5628:".
5624:og
5507:".
5478:doi
5442:".
5313:doi
5172:",
5071:doi
5048:23B
5042:".
4950:16B
4829:".
4798:".
4751:".
4739:, 2
4702:217
4494:= 7
4480:= 7
4473:= 5
4466:= 3
4433:= 6
4412:= 6
4388:= 6
4366:= 7
4356:= 7
4335:.
4307:= 5
4273:≡ 2
4260:≡ 2
4223:≡ 2
4062:= 5
3981:= 5
3969:= 5
3962:= 5
3955:= 5
3895:= 5
3880:xyz
3874:xyz
3865:= 5
3849:,
3820:= 0
3784:= 5
3750:= 5
3720:= 0
3437:= 3
3425:− 9
3401:+ 3
3259:+ 3
3251:= 6
3230:by
3162:= 3
3146:+ 3
3104:= 0
3079:+ 3
3064:− 3
3008:+ 3
2996:– 3
2974:+ 3
2966:− 3
2958:= 2
2954:= 2
2882:= 3
2870:− 9
2846:+ 3
2818:+ 3
2792:+ 3
2775:+ 3
2718:+ 3
2671:+ 3
2645:+ 3
2621:= (
2509:= −
2452:= 0
2417:= 4
2317:= 3
2266:'s
2230:= 3
2189:= 3
2178:= 3
2163:= 3
2150:by
2138:= 3
1998:= 2
1994:= 2
1951:= 2
1855:= 2
1839:= 2
1747:as
1706:= 2
1525:= 2
1406:= 2
1328:− 4
1309:+ 4
1283:− 4
1271:− 2
1261:+ 4
1249:+ 2
1207:= 4
1081:= 4
1070:= (
1045:= 4
999:= 4
964:xyz
952:xyz
553:If
517:GCD
431:of
311:= 0
267:= 4
142:.)
57:= 4
7054::
7005:.
6994:–
6964:.
6780:42
6778:.
6774:.
6730:,
6722:,
6706:,
6700:,
6533:MR
6531:,
6521:,
6453:98
6412:.
6386:.
6376:33
6374:.
6351:^
6337:.
6327:21
6325:.
6308:21
6306:.
6294:^
6286:15
6284:.
6267:97
6265:.
6229:.
6209:26
6201:cz
6197:by
6193:ax
6175:82
6173:.
6141:82
6139:.
6114:78
6112:.
6093:.
6076:.
6017:.
5983:.
5956:.
5917:37
5915:.
5911:.
5891:39
5889:.
5861:25
5859:.
5850:Az
5848:=
5844:+
5827:10
5825:.
5808:12
5806:.
5772:.
5766:az
5668:.
5634:34
5632:.
5620:,
5588:11
5586:.
5571:^
5526:.
5472:.
5468:.
5448:61
5446:.
5391:,
5377:.
5341:^
5327:.
5321:MR
5319:.
5309:82
5307:.
5254:^
5240:^
5220:^
5205:^
5121:71
5119:.
5077:.
5067:20
5065:.
5046:.
5017:39
4992:^
4984:23
4982:.
4948:.
4903:16
4901:.
4887:^
4857:.
4835:84
4833:.
4804:36
4802:.
4796:cz
4792:by
4788:ax
4757:18
4755:.
4727:,
4676:.
4627:^
4601:13
4599:.
4585:^
4572:10
4570:.
4549:,
4496:.
4475:,
4468:,
4414:,
4390:,
4256:+
4252:≡
4237:≡
4219:+
4215:≡
4201:≡
4165:,
4147:,
4132:+
4128:+
4111:≡
4099:≡
4087:≡
4079:,
3946:,
3942:,
3934:,
3930:,
3922:,
3918:,
3883:.
3857:.
3841:,
3837:,
3816:+
3812:+
3792:,
3736:,
3732:,
3716:+
3712:+
3699:=
3695:−
3684:=
3680:+
3669:=
3665:−2
3660:.
3652:,
3643:−
3633:+
3628:,
3603:−
3599:)(
3595:+
3539:−
3535:)(
3531:+
3519:−
3515:)(
3511:+
3499:−
3474:18
3445:−
3417:=
3397:=
3389:.
3372:+
3352:=
3348:+
3336:=
3332:18
3311:+
3300:18
3271:+
3267:(3
3255:(9
3221:.
3120:,
3116:,
3100:+
3096:+
3083:=
3068:=
3053:=
3049:−2
2990:,
2970:)(
2890:−
2862:=
2842:=
2814:=
2779:=
2764:=
2633:−
2625:+
2607:−
2603:=
2593:+
2589:=
2564:−
2560:=
2548:+
2544:=
2498:=
2460:,
2448:+
2444:+
2433:,
2429:,
2215:,
2211:,
2199:,
2124:,
2120:,
2108:,
2104:,
2077:,
2073:,
2059:=
2055:−
2051:=
2047:−
2043:=
2030:=
2020:=
2000:de
1981:+
1977:=
1966:−
1962:=
1953:de
1923:=
1919:+
1915:=
1901:+
1880:,
1876:,
1865:=
1841:uv
1792:−
1784:=
1763:+
1755:=
1702:+
1677:=
1673:−
1662:=
1658:+
1621:xy
1619:,
1615:,
1601:−
1597:=
1555:+
1551:=
1540:−
1536:=
1527:de
1511:,
1507:,
1470:−
1460:+
1431:−
1427:)(
1423:+
1411:.
1396:=
1391:,
1386:−
1382:=
1372:=
1368:−
1346:−
1320:−
1301:+
1279:=
1275:+
1273:uv
1257:=
1253:+
1251:uv
1236:=
1232:+
1216:uv
1214:−2
1205:uv
1190:=
1178:uv
1158:=
1154:+
1149:,
1132:uv
1119:,
1115:,
1101:.
1095:=
1091:+
1066:−
1056:=
1052:+
1032:=
1028:−
973:,
938:=
934:+
923:,
919:,
892:.
887:=
877:=
836:uv
834:=
757:,
753:,
742:=
738:+
727:,
723:,
714:.
657:,
653:,
638:=
634:+
615:=
608:+
601:=
597:=
593:+
579:,
575:,
563:,
559:,
544:,
536:,
528:,
505:,
501:,
492:gz
490:=
482:gy
480:=
475:,
472:gx
470:=
459:,
455:,
437:,
413:−
409:=
358:,
354:,
328:,
307:+
303:+
293:=
289:+
275:.
253:no
228::
216:,
212:,
203:gh
201:=
160:,
156:,
99:=
95:+
84:,
80:,
63:.
7042:.
7033:.
7020:.
7011:.
6998:.
6970:.
6952:.
6943:.
6921:.
6908:.
6896:.
6888:y
6884:x
6868:.
6845:.
6823:.
6790:.
6759:.
6718::
6691:.
6672:.
6646:.
6622:.
6598:.
6562:.
6488:.
6445:z
6441:y
6437:x
6414:9
6394:.
6382::
6368:n
6364:n
6345:.
6333::
6259:z
6255:y
6251:x
6233:.
6231:7
6167:z
6163:y
6159:x
6133:z
6129:y
6125:x
6101:.
6089::
6083:6
6072:z
6068:y
6064:x
6046:.
6036:n
6019:5
6011:z
6007:y
6003:x
5985:5
5977:z
5973:y
5969:x
5958:9
5929:.
5923::
5879:n
5846:y
5842:x
5800:C
5796:B
5792:A
5774:8
5762:y
5758:x
5706:.
5696:n
5670:8
5662:z
5658:y
5654:x
5626:C
5622:B
5618:A
5614:C
5610:B
5606:A
5528:6
5505:z
5501:y
5497:x
5484:.
5480::
5474:7
5440:y
5436:x
5417:z
5413:y
5409:x
5393:4
5379:6
5335:.
5315::
5292:.
5282:n
5153:n
5113:n
5085:.
5073::
5040:z
5036:y
5032:x
5009:n
5005:n
4942:z
4938:y
4934:x
4859:4
4827:z
4823:y
4819:x
4749:y
4745:x
4741:z
4737:y
4733:x
4729:z
4725:y
4721:x
4717:z
4678:5
4551:5
4522:.
4492:n
4485:n
4478:n
4471:n
4464:n
4457:n
4450:n
4431:n
4424:n
4417:n
4410:n
4400:n
4393:n
4386:n
4364:n
4354:n
4342:)
4338:(
4305:n
4289:x
4279:x
4275:x
4271:z
4269:−
4262:x
4258:y
4254:x
4250:z
4248:−
4239:y
4235:x
4225:x
4221:y
4217:x
4213:z
4211:−
4203:y
4199:x
4190:y
4184:x
4174:z
4168:y
4162:x
4156:z
4150:y
4144:x
4134:z
4130:y
4126:x
4113:z
4109:z
4101:y
4097:y
4089:x
4085:x
4071:θ
4060:n
3979:n
3967:n
3960:n
3953:n
3948:z
3944:y
3940:x
3936:z
3932:y
3928:x
3924:z
3920:y
3916:x
3908:.
3904:θ
3893:n
3863:n
3818:z
3814:y
3810:x
3801:z
3795:y
3789:x
3782:n
3763:.
3748:n
3740:)
3738:z
3734:y
3730:x
3728:(
3718:m
3714:l
3710:k
3701:m
3697:e
3693:f
3686:l
3682:f
3678:e
3671:k
3667:f
3658:m
3654:l
3650:k
3645:f
3641:e
3635:f
3631:e
3625:f
3623:2
3618:f
3612:e
3607:)
3605:f
3601:e
3597:f
3593:e
3591:(
3589:f
3587:2
3583:)
3578:3
3575:/
3571:r
3565:(
3561:r
3556:r
3551:3
3545:.
3543:)
3541:f
3537:e
3533:f
3529:e
3527:(
3525:f
3521:f
3517:e
3513:f
3509:e
3507:(
3505:f
3501:f
3497:e
3495:(
3493:f
3489:w
3485:r
3476:w
3469:v
3463:f
3457:e
3449:)
3447:f
3443:e
3441:(
3439:f
3435:w
3429:)
3427:f
3423:e
3421:(
3419:e
3415:v
3403:f
3399:e
3395:s
3386:f
3380:e
3374:v
3370:w
3368:3
3363:s
3354:s
3350:v
3346:w
3344:3
3338:r
3334:w
3325:s
3319:r
3313:v
3309:w
3307:3
3302:w
3295:v
3289:w
3283:v
3275:)
3273:v
3269:w
3265:w
3261:v
3257:w
3253:w
3249:z
3247:−
3239:z
3233:w
3227:u
3218:v
3212:w
3206:v
3200:v
3194:u
3188:w
3182:w
3176:u
3170:w
3164:w
3160:u
3154:u
3148:v
3144:u
3138:u
3136:2
3124:)
3122:z
3118:y
3114:x
3112:(
3102:m
3098:l
3094:k
3085:m
3081:f
3077:e
3070:l
3066:f
3062:e
3055:k
3051:e
3041:v
3035:u
3029:u
3023:e
3017:e
3012:)
3010:f
3006:e
3004:(
3000:)
2998:f
2994:e
2992:(
2987:e
2985:2
2978:)
2976:f
2972:e
2968:f
2964:e
2962:(
2960:e
2956:u
2952:r
2943:f
2937:e
2931:v
2925:u
2919:f
2913:e
2907:v
2901:u
2894:)
2892:f
2888:e
2886:(
2884:f
2880:v
2874:)
2872:f
2868:e
2866:(
2864:e
2860:u
2848:f
2844:e
2840:s
2832:f
2826:e
2820:v
2816:u
2812:s
2806:s
2800:s
2794:v
2790:u
2781:s
2777:v
2773:u
2766:r
2762:u
2760:2
2753:s
2747:r
2741:u
2735:z
2733:−
2720:v
2716:u
2710:u
2708:2
2703:v
2697:u
2691:v
2685:u
2679:z
2673:v
2669:u
2663:v
2657:u
2649:)
2647:v
2643:u
2641:(
2639:u
2635:v
2631:u
2627:v
2623:u
2619:z
2617:−
2609:v
2605:u
2601:y
2595:v
2591:u
2587:x
2581:v
2575:u
2566:y
2562:x
2558:v
2556:2
2550:y
2546:x
2542:u
2540:2
2532:y
2526:x
2517:x
2511:z
2507:x
2505:2
2500:y
2496:x
2490:y
2484:x
2475:z
2469:z
2463:y
2457:x
2450:z
2446:y
2442:x
2437:)
2435:z
2431:y
2427:x
2425:(
2415:n
2315:n
2228:n
2187:n
2176:n
2161:n
2154:.
2136:n
2128:)
2126:z
2122:y
2118:x
2116:(
2112:)
2110:k
2106:h
2102:g
2100:(
2095:x
2091:d
2087:g
2081:)
2079:k
2075:h
2071:g
2069:(
2061:k
2057:h
2053:g
2049:e
2045:d
2041:v
2032:h
2028:e
2022:g
2018:d
2012:e
2006:d
1996:m
1992:u
1983:e
1979:d
1975:x
1968:e
1964:d
1960:v
1949:u
1940:e
1934:d
1925:x
1921:v
1917:u
1910:2
1907:/
1903:t
1899:s
1884:)
1882:x
1878:v
1874:u
1872:(
1867:k
1863:v
1857:m
1853:u
1847:u
1837:y
1831:v
1825:u
1819:t
1813:s
1801:2
1798:/
1794:t
1790:s
1782:v
1772:2
1769:/
1765:t
1761:s
1753:u
1744:v
1738:u
1732:t
1726:s
1720:y
1714:x
1708:x
1704:t
1700:s
1694:t
1688:s
1679:t
1675:y
1671:x
1664:s
1660:y
1656:x
1647:z
1633:x
1629:d
1623:)
1617:e
1613:d
1611:(
1603:e
1599:d
1595:y
1592:x
1582:e
1578:d
1572:e
1566:d
1557:e
1553:d
1549:x
1542:e
1538:d
1534:y
1523:z
1515:)
1513:x
1509:z
1505:y
1503:(
1498:z
1492:y
1486:x
1472:y
1468:x
1462:y
1458:x
1452:y
1446:x
1437:z
1433:y
1429:x
1425:y
1421:x
1419:(
1408:s
1404:y
1398:w
1394:x
1388:v
1384:u
1380:z
1374:z
1370:y
1366:x
1359:.
1356:s
1352:w
1348:v
1344:u
1342:(
1333:.
1330:s
1326:w
1322:v
1318:u
1316:(
1311:s
1307:w
1303:v
1299:u
1297:(
1288:,
1285:s
1281:w
1277:v
1269:u
1263:s
1259:w
1255:v
1247:u
1238:w
1234:v
1230:u
1223:.
1220:s
1209:s
1203:2
1192:s
1185:2
1182:/
1166:s
1160:w
1156:v
1152:u
1139:2
1136:/
1123:)
1121:w
1117:v
1113:u
1111:(
1097:t
1093:s
1089:r
1079:n
1074:)
1072:a
1068:b
1064:c
1058:c
1054:b
1050:a
1043:n
1034:z
1030:y
1026:x
997:n
988:p
982:z
976:y
970:x
958:p
946:p
940:z
936:y
932:x
927:)
925:z
921:y
917:x
915:(
889:s
885:v
879:r
875:u
869:p
863:v
857:u
851:v
845:u
832:x
823:p
790:y
784:x
778:z
772:y
766:x
761:)
759:z
755:y
751:x
749:(
744:z
740:y
736:x
731:)
729:z
725:y
721:x
719:(
712:x
708:x
669:g
661:)
659:z
655:y
651:x
649:(
643:.
640:z
636:y
632:x
620:z
617:g
613:y
610:g
606:x
603:g
599:c
595:b
591:a
583:)
581:z
577:y
573:x
571:(
567:)
565:c
561:b
557:a
555:(
546:z
542:y
538:z
534:x
530:y
526:x
509:)
507:z
503:y
499:x
497:(
488:c
478:b
468:a
463:)
461:c
457:b
453:a
451:(
446:c
440:b
434:a
424:g
415:a
411:c
407:b
398:b
391:d
385:c
379:a
373:d
367:d
362:)
360:c
356:b
352:a
350:(
337:c
331:b
325:a
309:c
305:b
301:a
295:c
291:b
287:a
281:p
272:p
265:n
258:n
247:.
245:)
243:c
239:b
235:a
233:(
225:h
220:)
218:c
214:b
210:a
208:(
199:n
193:g
187:n
181:h
175:n
169:n
164:)
162:c
158:b
154:a
152:(
135:n
129:b
123:a
113:n
107:n
101:c
97:b
93:a
88:)
86:c
82:b
78:a
76:(
55:n
49:n
43:n
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
