Pierre de Fermat gave a criterion for numbers of the form 8a + 1 and 8a + 3 to be sums of a square plus twice another square, but did not provide a proof.[1] N. Beguelin noticed in 1774[2] that every positive integer which is neither of the form 8n + 7, nor of the form 4n, is the sum of three squares, but did not provide a satisfactory proof.[3] In 1796 Gauss proved his Eureka theorem that every positive integer n is the sum of 3 triangular numbers; this is equivalent to the fact that 8n + 3 is a sum of three squares. In 1797 or 1798 A.-M. Legendre obtained the first proof of his 3 square theorem.[4] In 1813, A. L. Cauchy noted[5] that Legendre's theorem is equivalent to the statement in the introduction above. Previously, in 1801, C. F. Gauss had obtained a more general result,[6] containing Legendre's theorem of 1797–8 as a corollary. In particular, Gauss counted the number of solutions of the expression of an integer as a sum of three squares, and this is a generalisation of yet another result of Legendre,[7] whose proof is incomplete. This last fact appears to be the reason for later incorrect claims according to which Legendre's proof of the three-square theorem was defective and had to be completed by Gauss.[8]
The "only if" of the theorem is simply because modulo 8, every square is congruent to 0, 1 or 4. There are several proofs of the converse (besides Legendre's proof). One of them is due to J. P. G. L. Dirichlet in 1850, and has become classical.[9] It requires three main lemmas:
the equivalence class of the trivial ternary quadratic form.
Relationship to the four-square theoremedit
This theorem can be used to prove Lagrange's four-square theorem, which states that all natural numbers can be written as a sum of four squares. Gauss[10] pointed out that the four squares theorem follows easily from the fact that any positive integer that is 1 or 2 mod 4 is a sum of 3 squares, because any positive integer not divisible by 4 can be reduced to this form by subtracting 0 or 1 from it. However, proving the three-square theorem is considerably more difficult than a direct proof of the four-square theorem that does not use the three-square theorem. Indeed, the four-square theorem was proved earlier, in 1770.
^A.-M. Legendre, Hist. et Mém. Acad. Roy. Sci. Paris, 1785, pp. 514–515.
^See for instance: Elena Deza and M. Deza. Figurate numbers. World Scientific 2011, p. 314 [1]
^See for instance vol. I, parts I, II and III of : E. Landau, Vorlesungen über Zahlentheorie, New York, Chelsea, 1927. Second edition translated into English by Jacob E. Goodman, Providence RH, Chelsea, 1958.
^Gauss, Carl Friedrich (1965), Disquisitiones Arithmeticae, Yale University Press, p. 342, section 293, ISBN0-300-09473-6
December 15, 2023
legendre, three, square, theorem, mathematics, states, that, natural, number, represented, three, squares, integers, displaystyle, only, form, displaystyle, nonnegative, integers, distances, between, vertices, double, unit, cube, square, roots, first, natural,. In mathematics Legendre s three square theorem states that a natural number can be represented as the sum of three squares of integers n x 2 y 2 z 2 displaystyle n x 2 y 2 z 2 if and only if n is not of the form n 4 a 8 b 7 displaystyle n 4 a 8b 7 for nonnegative integers a and b Distances between vertices of a double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem 7 is not possible due to Legendre s three square theorem The first numbers that cannot be expressed as the sum of three squares i e numbers that can be expressed as n 4 a 8 b 7 displaystyle n 4 a 8b 7 are 7 15 23 28 31 39 47 55 60 63 71 sequence A004215 in the OEIS ab 0 1 20 7 28 1121 15 60 2402 23 92 3683 31 124 4964 39 156 6245 47 188 7526 55 220 8807 63 252 10088 71 284 11369 79 316 126410 87 348 139211 95 380 152012 103 412 1648Unexpressible valuesup to 100 are in boldContents 1 History 2 Proofs 3 Relationship to the four square theorem 4 See also 5 NotesHistory editPierre de Fermat gave a criterion for numbers of the form 8a 1 and 8a 3 to be sums of a square plus twice another square but did not provide a proof 1 N Beguelin noticed in 1774 2 that every positive integer which is neither of the form 8n 7 nor of the form 4n is the sum of three squares but did not provide a satisfactory proof 3 In 1796 Gauss proved his Eureka theorem that every positive integer n is the sum of 3 triangular numbers this is equivalent to the fact that 8n 3 is a sum of three squares In 1797 or 1798 A M Legendre obtained the first proof of his 3 square theorem 4 In 1813 A L Cauchy noted 5 that Legendre s theorem is equivalent to the statement in the introduction above Previously in 1801 C F Gauss had obtained a more general result 6 containing Legendre s theorem of 1797 8 as a corollary In particular Gauss counted the number of solutions of the expression of an integer as a sum of three squares and this is a generalisation of yet another result of Legendre 7 whose proof is incomplete This last fact appears to be the reason for later incorrect claims according to which Legendre s proof of the three square theorem was defective and had to be completed by Gauss 8 With Lagrange s four square theorem and the two square theorem of Girard Fermat and Euler the Waring s problem for k 2 is entirely solved Proofs editThe only if of the theorem is simply because modulo 8 every square is congruent to 0 1 or 4 There are several proofs of the converse besides Legendre s proof One of them is due to J P G L Dirichlet in 1850 and has become classical 9 It requires three main lemmas the quadratic reciprocity law Dirichlet s theorem on arithmetic progressions and the equivalence class of the trivial ternary quadratic form Relationship to the four square theorem editThis theorem can be used to prove Lagrange s four square theorem which states that all natural numbers can be written as a sum of four squares Gauss 10 pointed out that the four squares theorem follows easily from the fact that any positive integer that is 1 or 2 mod 4 is a sum of 3 squares because any positive integer not divisible by 4 can be reduced to this form by subtracting 0 or 1 from it However proving the three square theorem is considerably more difficult than a direct proof of the four square theorem that does not use the three square theorem Indeed the four square theorem was proved earlier in 1770 See also editFermat s two square theorem Sum of two squares theoremNotes edit Fermat to Pascal PDF September 25 1654 Archived PDF from the original on July 5 2017 Nouveaux Memoires de l Academie de Berlin 1774 publ 1776 pp 313 369 Leonard Eugene Dickson History of the theory of numbers vol II p 15 Carnegie Institute of Washington 1919 AMS Chelsea Publ 1992 reprint A M Legendre Essai sur la theorie des nombres Paris An VI 1797 1798 p 202 and pp 398 399 A L Cauchy Mem Sci Math Phys de l Institut de France 1 14 1813 1815 177 C F Gauss Disquisitiones Arithmeticae Art 291 et 292 A M Legendre Hist et Mem Acad Roy Sci Paris 1785 pp 514 515 See for instance Elena Deza and M Deza Figurate numbers World Scientific 2011 p 314 1 See for instance vol I parts I II and III of E Landau Vorlesungen uber Zahlentheorie New York Chelsea 1927 Second edition translated into English by Jacob E Goodman Providence RH Chelsea 1958 Gauss Carl Friedrich 1965 Disquisitiones Arithmeticae Yale University Press p 342 section 293 ISBN 0 300 09473 6 Retrieved from https en wikipedia org w index php title Legendre 27s three square theorem amp oldid 1164769990, wikipedia, wiki, book, books, library,
