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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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    $\begingroup$ Actually, for 3-term arithmetic progressions, this was proved much earlier by van der Corput. $\endgroup$ – JSE Mar 5 '13 at 16:00
  • $\begingroup$ 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 $k \ll p^{1-\epsilon}$? $\endgroup$ – Xander Faber Mar 5 '13 at 17:33
  • $\begingroup$ @JSE: Sorry for the wrong attribution. I should have looked it up first. $\endgroup$ – Michael Stoll Mar 5 '13 at 20:17
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To add to what Michael says: from the newer theorem of Green, Tao, and Ziegler

http://arxiv.org/abs/1009.3998

one knows that all three-term homogeneous linear equations have infinitely many solutions, not only in prime powers, but in primes; that the number of such solutions is asymptotically exactly what you expect; and that the same statement holds under mild conditions for systems of linear equations as well.

(It may be that the specific case you ask about is handled by older methods a la van der Corput, but I don't know offhand.)

It makes your question much harder if you choose a prime when you wake up in the morning and demand that X be a power of THAT prime. I don't know, for instance, whether anyone has proven the statement you mention, that there are infinitely many powers of 2 which are the sum of two primes.

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