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Math 2.0

Number theory Diophantine equations Theorem

Legendre's three-square theorem

Statement

Let $n$ be a natural number. Then $n = x^2 + y^2 + z^2$ for some integers $x$ , $y$ and $z$ if and only if $n \neq 4^a (8b + 7)$ for all natural numbers $a$ and $b$ .

Remarks

The "only if" part is easy: a square is $0$ , $1$ or $4$ modulo $8$ , so no sum of three squares is $7$ modulo $8$ . The first numbers that are not sums of three squares are $7$ , $15$ , $23$ , $28$ and $31$ . The theorem implies Lagrange's four-square theorem in a few lines.

History

Legendre gave the first proof in 1797 or 1798. Gauss proved a more general result in 1801.

Further reading

Web https://en.wikipedia.org/wiki/Legendre%27s_three-square_theorem, lead section
Web https://en.wikipedia.org/wiki/Legendre%27s_three-square_theorem, History

About this page

Tag	`0B64Q7XNE3`
URI	`m2://0B64Q7XNE3`
Kind	statement, theorem
Writing	W0 stub
MSC	11E25
Status	This copy of the site has no build data. Run `m2 status 0B64Q7XNE3`.
Same as	Wikidata Q15940470 Wikipedia Legendre's three-square theorem

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