A Diophantine equation involving prime powers. Dear all,
When working with a group theory problem, I come up with the equation:
$$p^x-1=2^y,$$ where $p$ is a prime and $x,y$ are positive integers. I would like to show that this equation has infinitely many solutions $(p,x,y)$ ($p$ is also a variable!) but not sure if this is correct. So my question is:
Is it correct? If yes, how do we prove it?
I have another question: 
Are there infinitely many primes of the form $a2^n+1$, where $a$ is a fixed number? For example, just take $a=3$.
 A: I only answer the (newly) added question (the others being adressed in comments): 

Are there infinitely many primes of the form $a2^n + 1$, where $a$ is a fixed number?

Certainly not for each $a$. More precisely, Sierpiński (1060) showed that there exist infinitely many odd $a$ such that all numbers in the set 
$$
\lbrace a2^n +1 \colon n \in \mathbb{N} \rbrace
$$ 
are composite. 
Such an $a$ is called a Sierpiński Number; an explict example is $78557$. Chances are this is the smallest example, and there is some ongoing computing effort to show this. See the link I gave above for further details. 
For certain other $a$ there are likely infinitely many, but this is never known. The point is that the most naive heuristic would be to say that the probability of $a2^n+1$ to be prime is proportional to $1/n$ (more precisely $1/\log (a2^n +1)$ by the Prime Number Theorem) and the series over $1/n$ being divergent one expects infinitely many, just like for Mersenne Primes. 
However, and necessarily in view of what I said above, there can be problems with this heuristic: Depending on the $a$ there can be 'local' restrictions (that is one finds congruences that impede the number of this form to be prime, see again the site above).
Or/and, as in the Fermat case, there is a general factorization that reduces the range of the admissible exponents so much that the relevant series will converge and one thus expects at most finitely many. 
One more related key-word: Primes of the form $a2^n + 1$ for $a$ odd not fixed, but $2^n \gt a$ (to avoid trivializing the condition) are called Proth primes. 
Following the links on the two pages I gave you will find some more related notions and additional information. 
