Legendre's three-square theorem

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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

n=x^{2}+y^{2}+z^{2}\

if and only if n is not of the form $n=4^{a}(8b+7)$ for integers a and b.

The first numbers that cannot be expressed as the sum of three squares (i.e. numbers that can be expressed as $n=4^{a}(8b+7)$ ) are

7, 15, 23, 28, 31, 39, 47, 55, 60, 63, 71 ... (sequence A004215 in the OEIS).

History

N. Beguelin notices in 1774 that every positive integer which is neither of the form 8n + 7, nor of the form 4n, is the sum of three squares, but doesn't provide a satisfactory proof. In 1797 or 1798 A.-M. Legendre obtains the first proof of this assertion. In 1813, A. L. Cauchy notes 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, 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, whose proof is incomplete. This last fact appears to be the reason of later incorrect claims according to which Legendre's proof of the three-square theorem was defective and had to be completed by Gauss.

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

The part « only if » of the theorem is simply due to the fact that modulo 8, every square is congruent to 0, 1 or 4. There are several proofs of the reciprocal. One of them is due to J. P. G. L. Dirichlet in 1850, and has become classical. 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

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. Let n be a natural number, then there are two cases:

either n is not of the form $4^{a}(8b+7)$ , in which case it is a sum of three squares and thus of four squares $n=x^{2}+y^{2}+z^{2}+0$ for some x, y, z, by Legendre–Gauss;
or $n=4^{a}(8b+7)=(2^{a})^{2}((8b+6)+1)$ , where $8b+6=2(4b+3)$ , which is again a sum of three squares by Legendre–Gauss, so that n is a sum of four squares.

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.

Notes

Nouveaux Mémoires de l'Académie de Berlin (1774, publ. 1776), p. 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 théorie des nombres, Paris, An VI (1797-1798), p. 202 et 398-399.
A. L. Cauchy, Mém. 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 Mém. Acad. Roy. Sci. Paris, 1785, p. 514-515.
See for instance: Elena Deza and M. Deza. Figurate numbers. World Scientific 2011, p.314
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.
France Dacar (2012). "The three squares theorem & enchanted walks" (PDF). Jozef Stefan Institute. Retrieved 6 October 2013.

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