The question in interest is solutions to the Diophantine equation $p^x-q^y=p-q$ where p,q are prime and x,y are positive integers.
I found a math overflow question (2014) where this problem was considered: https://mathoverflow.net/questions/158071/integer-solutions-of-xyn-y-xm-for-n-m. Here the consensus was that the problem is still open (not sure if they were only referring to the case where $p,q$ doesn't have to be odd primes).
I also found a paper in the journal of mathematics by Scott R. (1993) way before the math-overflow question was posted: https://www.sciencedirect.com/science/article/pii/S0022314X83710413. The paper has a complete proof to the question addressed.
There is also another paper from Mignonette and Petho (1999) where they completely solve a sub case of the problem: https://www.jstor.org/stable/43736656
Looking at the dates of publications the story being said is a little confusing and I may not be understanding things completely and clarification on the standing of this problem will be greatly helpful.