Consider the integers $\alpha,\beta,\gamma,n>0$.
In which cases does the relation
$$ \gamma^n=\sum_{k=1}^{n-1}\binom{n}{k}\alpha^{n-k}\beta^k $$$$ \gamma^n=\sum_{k=1}^{n-1}\binom{n}{k}\alpha^{n-k}\beta^k=(\alpha+\beta)^n-\alpha^n-\beta^n $$
hold?
The problem rises in the context of Waring's formula (link in Italian, but readable). In fact, since
$$ (\alpha+\beta)^{n}-\alpha^{n}-\beta^{n}=\sum_{k=1}^{f_{1}}T_{k}*\alpha^{k}\beta^{k}*[\alpha^{n-2k}+\beta^{n-2k}]-f_{2} $$
it makes sense to ask if the above sum (of mixed terms) may equal to the exact $n$-power of a third integer $\gamma$.