Legendre's three-square theorem

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 nonnegative 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).

a b	0	1	2
0	7	28	112
1	15	60	240
2	23	92	368
3	31	124	496
4	39	156	624
5	47	188	752
6	55	220	880
7	63	252	1008
8	71	284	1136
9	79	316	1264
10	87	348	1392
11	95	380	1520
12	103	412	1648
Unexpressible values up to 100 are in bold

History

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]

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 "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

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.

Notes

^ "Fermat to Pascal" (PDF). September 25, 1654. Archived (PDF) from the original on July 5, 2017.
^ Nouveaux Mémoires de l'Académie 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 théorie des nombres, Paris, An VI (1797–1798), p. 202 and pp. 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, 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, ISBN 0-300-09473-6