Power of an integer as exact sum of mixed terms

Question

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 (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 $\gamma$.

Answer 1

Euler in 1769 conjectured 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.