Legendre's three-square theorem

In mathematics, Legendre's three-square theorem states that any natural number that is not of the form $n=4^{a}(8b+7)$ for integers a and b can be represented as the sum of three integer squares:

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

This theorem was stated by Adrien-Marie Legendre in 1798. His proof was incomplete, leaving a gap which was later filled by Carl Friedrich Gauss.

The first numbers that cannot be expressed as the sum of three squares (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).

Relationship to the Four-Square Theorem

This theorem leads to an easy proof of 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.

See also

• Fermat's two-square theorem
• Lagrange's four-square theorem