# Catalan-type equations for prime powers

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?

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It is known by results of Green and Tao that there are infinitely many 3-term sequences of primes in arithmetic progression. So taking $a = b = 1$ and $c = 2$, you have infinitely many solutions in primes.

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Actually, for 3-term arithmetic progressions, this was proved much earlier by van der Corput. –  JSE Mar 5 '13 at 16:00
@Michael and JSE: Many thanks! –  Xander Faber Mar 5 '13 at 17:27
Suppose that $p,p+k,p+2k$ is a $3$-term arithmetic progression of primes with $p$ large. Does one have any control over $k$? For example, is it possible to take $k \ll p^{1-\epsilon}$? –  Xander Faber Mar 5 '13 at 17:33
@JSE: Sorry for the wrong attribution. I should have looked it up first. –  Michael Stoll Mar 5 '13 at 20:17