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In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers : if and only if is not of the form for integers and . The first numbers that cannot be expressed as the sum of three squares (i.e. numbers that can be expressed as ) are :7, 15, 23, 28, 31, 39, 47, 55, 60, 63, 71 ... . == History == Pierre de Fermat gave a criterion for numbers of the form 3''a''+1 to be a sum of 3 squares essentially equivalent to Legendre's theorem, but did not provide a proof. N. Beguelin noticed in 1774〔''Nouveaux Mémoires de l'Académie de Berlin'' (1774, publ. 1776), .〕 that every positive integer which is neither of the form 8''n'' + 7, nor of the form 4''n'', is the sum of three squares, but did not provide a satisfactory proof.〔Leonard Eugene Dickson, History of the theory of numbers, vol. II, p. 15 (Carnegie Institute of Washington 1919; AMS Chelsea Publ., 1992, reprint).〕 In 1796 Gauss proved his Eureka theorem that every positive integer ''n'' is the sum of 3 triangular numbers; this is trivially equivalent to the fact that 8''n''+3 is a sum of 3 squares. In 1797 or 1798 A.-M. Legendre obtains the first proof of his 3 square theorem.〔A.-M. Legendre, ''Essai sur la théorie des nombres'', Paris, An VI (1797-1798), .〕 In 1813, A. L. Cauchy notes〔A. L. Cauchy, ''Mém. Sci. Math. Phys. de l'Institut de France'', (1) 14 (1813-1815), 177.〕 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,〔C. F. Gauss, ''Disquisitiones Arithmeticae'', Art. 291 et 292.〕 containing Legendre theorem of 1797-8 as a corollary. In particular, Gauss counts 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,〔A.-M. Legendre, ''Hist. et Mém. Acad. Roy. Sci. Paris'', 1785, .〕 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.〔See for instance: Elena Deza and M. Deza. Figurate numbers. World Scientific 2011, p.314 ()〕 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Legendre's three-square theorem」の詳細全文を読む スポンサード リンク
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