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

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

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

See also

• Fermat's two-square theorem
• Sum of two squares theorem
• Legendre's equation