# Question on Sums of Squares

Is it possible for two different $n$-element sets, each of which consists of $n$ unique positive integers (they can appear in both sets, though) to have the same sum when the squares of their elements are added?

Edit: For obvious reasons, I'm not considering the case $n=1$.

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$8,1$ and $7,4$ – Will Jagy Jul 24 '12 at 20:07
This is not really an appropriate question for the site (see the FAQ). Try math.stackexchange.com instead. – Qiaochu Yuan Jul 24 '12 at 20:12

Yes. One way to see this is that there are more $n$-element subsets with terms up to $N$ than there are possible sums of squares, giving an answer by the pigeonhole principle.
A more beautiful answer was given by Prouhet in the 1850's, who exhibited for each $n$ an explicitly-defined pair of sets $A$ and $B$ of size $2^n$ such that $$\sum_{a\in A}a^k=\sum_{b\in B}b^k\text{ for each 1\le k\le n}.$$
If you include 0, there are other interesting families of solutions: $3^2+4^2=0^2+5^2$, $10^2+11^2+12^2=0^2+13^2+14^2$, $21^2+22^2+23^2+24^2 = 0^2 + 25^2+26^2+27^2$, and so on. The pattern is left to the reader. – Kevin O'Bryant Jul 25 '12 at 1:07