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How can I prove that every integer $n>=170$ can be written as a sum of five positive squares? (i.e. none of the squares are allowed to be zero). I know that $169=13^2=12^2+5^2=12^2+4^2+3^2=10^2+8^2+2^2+1^2$, and $n-169=a^2+b^2+c^2+d^2$ for some integers $a$, $b$, $c$, $d$, but do I show it? Thank you.

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up vote 11 down vote accepted

Hint: let $n-169 = a^2+b^2+c^2+d^2$; if $a,b,c,d \neq 0$ then ... if $d = 0$ and $a,b,c \neq 0$ then ... if $c = d = 0$ and $a,b \neq 0$ then ... if $b = c = d = 0$ and $a \neq 0$ then ... if $a = b = c = d = 0$ then - wait, that can't happen!

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@kira: remember what you know about $169$: it can be written as a square, a sum of two squares, a sum of three squares and a sum of four squares! – t.b. Apr 10 '11 at 0:20
@Theo: Good point. Thanks! – kira Apr 10 '11 at 0:31
Thanks! – kira Apr 10 '11 at 0:33

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