Can you prove the following:

Conjecture:  Let $\ p\in\mathbb P\ $ be an arbitrary prime. Then there exist two relatively prime integers $\ a\ $ and $\ b\ $ such that $\ a>0\ $ and $\ b>1\ $ and

$$ \frac{b^p-a^p}{b-a} \in\ \mathbb P. $$

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A harder question would be a restriction to $\ b=a+1.$

See also the comment below by @GHfromMO.

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Finally, one may ask the same question about $\ 2\, <\, p\, \in\,\mathbb P\ $ but for:

$$ \frac {a^p+b^p}{a+b}. $$