I have the following question:
- Do there exist two (non-empty) sets of natural numbers (each one containing the same number of elements $N$), {$m_1,\ldots m_N$} and {$n_1,\ldots n_N$} such that all $m$'s and all $n$'s are different and $m_1+\ldots +m_N=n_1+\ldots +n_N$ and $m_1^2+\ldots +m_N^2=n_1^2+\ldots+n_N^2$?
Do there exist two sets of natural numbers (each one containing the same amount of elements $N$), {$m_1,\ldots m_N$} and {$n_1,\ldots n_N$} such that all $m$'s and all $n$'s are different and $m_1+\ldots +m_N=n_1+\ldots +n_N$ and $m_1^2+\ldots +m_N^2=n_1^2+\ldots+n_N^2$? II don't see any counterexample, but assuming that there does exist one, can we add more similar polynomial identities, maybe qubes or something to prevent such counterexamples?
