Lagrange's four-square theorem

Lagrange's four-square theorem, also known as Bachet's Conjecture, was proved in 1770 by Joseph Louis Lagrange.

It states that every positive integer can be expressed as the sum of four squares of integers.

More formally, for every positive integer n there exist non-negative integers a,b,c,d such that

n = a² + b² + c² + d².

Adrien-Marie Legendre improved on the theorem in 1798 by stating that a positive integer can be expressed as the sum of three squares iff it is not of the form 4^k(8m + 7). His proof was incomplete, leaving a gap which was later filled by Carl Friedrich Gauss.

Lagrange's four-square theorem is a special case of the Fermat polygonal number theorem and Waring's problem.

In 2005, Zhi-Wei Sun proved that any natural number can be represented as the sum of a square, an even square and a triangular number.

External links

• Proof at PlanetMath.org
• Another proof

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