Is there a fixed positive integer $ a$ for which $ a(b+c) | a^b+a^c$ with $b, c$ are positive integers?

Question

I have tried many times to find a closed form of $a$ for which the below question satisfied , Now my question is how do i can proof or disproof the existence of the fixed positive integer $a$ with $b, c$ are also positive integers

Question: Is there a fixed positive integer $ a$ for which $ a(b+c) | a^b+a^c$ with $b, c$ are positive integers ?

Answer 1

Suppose that such an $a$ exists. Then, if $p$ is a prime with $p>a$, for $b=c=p$ we have that $$2ap|2a^p\Rightarrow p|a^{p-1},$$ hence $p|a$, which is a contradiction.