---
tag: 0B64Q7XNE3
kind: statement
role: theorem
name: Legendre's three-square theorem
msc: ["11E25"]
sources:
  - url: https://en.wikipedia.org/wiki/Legendre%27s_three-square_theorem
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Legendre%27s_three-square_theorem
    locator: History
same_as:
  wikidata: Q15940470
  wikipedia: 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|0KYYSMEVHH]] in a few lines.

## History

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

## Explanations

### L1 human

Most numbers are sums of three squares, for example $6 = 4 + 1 + 1$.
The exceptions are $7, 15, 23, \dots$, the numbers that leave remainder $7$ on division by $8$, and these numbers times powers of $4$.