Minimal value of a diophantine expression

Question

Given the expression $$x^a - y^b>0$$ what is the minimum positive value it can have given $x>y > 1$ and $a,b>1$. For example, if I have $4^a - 3^b$ I would conjecture that the smallest value would be for $4^2 - 3^2 = 7$. Another close solution would be $4^4 - 3^5 = 13$. But how would I formally prove it, and if there is a smaller value how would I find the smallest one?

Answer 1

For $4^a-3^b$, by Mihăilescu's theorem (in reference to solutions of the equation $x^a - y^b = 1$) we know that $8,9$ are the only consecutive powers and so $4^a-3^b > 1$. Working $\mod 2,3$ it is clear that $4^a-3^b \neq 2,3,4,6$. Rechecking $\mod 3$ we have for the equation $4^a-3^b = 5$ that $1 \equiv 2 \mod 3$ meaning that $5$ is not a solution either and $7 = 4^a - 3^b$ is the minimal solution.