I am to find all possible values of a and b where a, b are positive

Question

integers and to show that I have found all of them.For example:
"2^3 = 3^1 + 5" is a satisfied equation where(a, b) are (3,1).
I know that it can be solved using modular arithmetic but I am NEW to the concept so I would honestly appreciate if you can explain me the same example with the most basic steps in modular arithmetic.Awaiting your reply.Thank you.