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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 $\gamma$.

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

In which cases does the relation

$$ \gamma^n=\sum_{k=1}^{n-1}\binom{n}{k}\alpha^{n-k}\beta^k $$

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 $\gamma$.

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 $\gamma$.

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Power of an integer as exact sum of mixed terms

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

In which cases does the relation

$$ \gamma^n=\sum_{k=1}^{n-1}\binom{n}{k}\alpha^{n-k}\beta^k $$

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 $\gamma$.