So I'm aware of the Pillai's conjecture that says that $Ax^n-By^m=C$ has only finitely many solutions $(x,y,m,n)$ if $(m,n)\neq (1,1)$, when $A,B,C$ are fixed positive integers. I have the following questions:
- Is it also a conjecture that $Ap-Bq=C$ (where $p,q$ are prime numbers) has infinitely many solutions $(p,q)$? (I believe this should be true, this reminds me of the Twin prime conjecture.)
- Is is also a conjecture that $Ax^n+By^m=C$ has only finitely many solutions $(x,y,m,n)$ if $(m,n)\neq (1,1)$? (The Pillai's conjecture states that $A$ and $B$ are positive integers, I'm not really sure why.)
- Is is also a conjecture that $Ap+Bq=C$ (where $p,q$ are prime numbers) has infinitely many solutions $(p,q)$?
I'm familiar with Generalized Tijdeman's problem as well, but I'm not sure how those relate, since it doesn't have the factors $A,B$ and $C.$