Editing Lagrange's four-square theorem (section)

==Number of representations==
{{main|Jacobi's four-square theorem}}
The number of representations of a natural number ''n'' as the sum of four squares of integers is denoted by ''r''<sub>4</sub>(''n''). [[Jacobi's four-square theorem]] states that this is eight times the sum of the [[divisor]]s of ''n'' if ''n'' is odd and 24 times the sum of the odd divisors of ''n'' if ''n'' is even (see [[divisor function]]), i.e.

<math display="block">r_4(n)=\begin{cases}8\sum\limits_{m\mid n}m&\text{if }n\text{ is odd}\\[12pt]
24\sum\limits_{\begin{smallmatrix} m|n \\ m\text{ odd} \end{smallmatrix}}m&\text{if }n\text{ is even}.
\end{cases}</math>

Equivalently, it is eight times the sum of all its divisors which are not divisible by 4, i.e.

<math display="block">r_4(n)=8\sum_{m\,:\, 4\nmid m\mid n}m.</math>

We may also write this as
<math display="block">r_4(n) = 8 \sigma(n) -32 \sigma(n/4) \ , </math>
where the second term is to be taken as zero if ''n'' is not divisible by 4. In particular, for a [[prime number]] ''p'' we have the explicit formula&nbsp;{{math|1=''r''<sub>4</sub>(''p'') = 8(''p'' + 1)}}.<ref name="Williams_2011">{{harvnb|Williams|2011|p=119}}.</ref>

Some values of ''r''<sub>4</sub>(''n'') occur infinitely often as {{math|1=''r''<sub>4</sub>(''n'') = ''r''<sub>4</sub>(2<sup>''m''</sup>''n'')}} whenever ''n'' is even. The values of ''r''<sub>4</sub>(''n'')/''n'' can be arbitrarily large: indeed, ''r''<sub>4</sub>(''n'')/''n'' is infinitely often larger than 8{{radic|log ''n''}}.<ref name="Williams_2011" />