Number theory Diophantine equations Theorem

Lagrange's four-square theorem

Statement

Each natural number $n$ is a sum of four squares of integers: $n = a^2 + b^2 + c^2 + d^2$ for some integers $a$ , $b$ , $c$ and $d$ .

Remarks

For example, $7 = 4 + 1 + 1 + 1$ and $15 = 9 + 4 + 1 + 1$ . Three squares are not enough: $7$ is not a sum of three squares, by Legendre's three-square theorem. By Euler's four-square identity, it is enough to prove the theorem for primes. Other proofs use Minkowski's theorem or the Hurwitz quaternions.

History

Diophantus knew examples, and Bachet stated the theorem in his Latin translation of the Arithmetica of 1621. Lagrange gave the first proof in 1770.

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Euler's four-square identity statement
Lagrange's four-square theorem statement

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Tag	`0KYYSMEVHH`
URI	`m2://0KYYSMEVHH`
Kind	statement, theorem
Writing	W1 stated
MSC	11E25
Status	This copy of the site has no build data. Run `m2 status 0KYYSMEVHH`.
Uses	Euler's four-square identity
Same as	Wikidata Q756946 Wikipedia Lagrange's four-square theorem

Lagrange's four-square theorem

Statement

Remarks

History

Reading path

Where this leads

Further reading

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