Do there exist nonzero integers $a,b,c$ for which the equation $$aX + bY = cZ$$ has infinitely many solutions with $X,Y,Z$ distinct prime powers?
For example, if there are infinitely many Sophie Germain primes $p$ (so that $2p+1$ is also prime), then we can take $a = 2$, $b = c = 1$. Triples of the form $(X,Y,Z) = (p, 1, 2p+1)$ satisfy the equation.
For another example, if there are infinitely many Mersenne primes $p = 2^n - 1$, then we could take $a = c = 1$, $b = -1$ and look at triples of the form $(X,Y,Z) = (2^n, 1, p)$.
Or if the twin prime conjecture is true, then one can take $a = c = 1$ and $b = 2$ and arrive at infinitely many solutions of the form $(X,Y,Z) = (p, 1, p+2)$ with $p$ prime.
Or if the Goldbach conjecture is true, then take $a = b = c = 1$. For each $n \geq 1$, a decomposition of $2^n$ as $p+q$ gives a solution $(X,Y,Z) = (p,q,2^n)$.
(Note that one must require the prime powers $X,Y,Z$ to be distinct in order to rule out trivial solutions like $p^n + p^n = 2p^n$.)
But what I really want to know is: do there exist any examples of an equation of the above sort for which an unconditional answer is known?
