Consider the integers $\alpha,\beta,\gamma,n>0$. 


In which cases does the relation



> 
> $$
> \gamma^n=\sum\_{k=1}^{n-1}\binom{n}{k}\alpha^{n-k}\beta^k=(\alpha+\beta)^n-\alpha^n-\beta^n
> $$
> 
> 
> 


hold? 


The problem rises in the context of [Waring's formula](https://it.wikipedia.org/wiki/Formule_di_Waring) (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 $\gamma$.



Power of an integer as exact sum of mixed terms
Euler in 1769 [conjectured](https://en.m.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture) that for all integers n and k greater than 1, if the sum of k nth powers of positive integers is itself a nth power, then k is greater than or equal to n:


$$a^n\_1 + a^n\_2 + ... + a^n\_k = b^n ⇒ k ≥ n$$


The conjecture holds for the case n = 3; it was disproved for n = 4 and n = 5. It is unknown whether the conjecture fails or holds for any value n ≥ 6.


Your equation has $k=3$. Thus **Euler's conjecture for $n=6$ implies** that your equation has no solution with n ≥ 6.


