Question on Sums of Squares - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T08:05:03Z http://mathoverflow.net/feeds/question/103044 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103044/question-on-sums-of-squares Question on Sums of Squares rebeccakuang 2012-07-24T20:03:00Z 2012-07-24T20:09:12Z <p>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?</p> <p>Edit: For obvious reasons, I'm not considering the case $n=1$.</p> http://mathoverflow.net/questions/103044/question-on-sums-of-squares/103045#103045 Answer by Anthony Quas for Question on Sums of Squares Anthony Quas 2012-07-24T20:09:12Z 2012-07-24T20:09:12Z <p>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.</p> <p>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 $$\sum_{a\in A}a^k=\sum_{b\in B}b^k\text{ for each 1\le k\le n}.$$</p>