How can we generate a $2$-digit number $XY$ on base $B$, such that $BX+Y=Y^X$?

Question

For example, $25$ on base $10$ is equal to $5^2$.

This should be pretty easy to solve using fairly simple arithmetic.

But I'm finding it hard to generate any other solutions besides the one mentioned above.

Answer 1

Here are two more specific solutions: $52_6=2^5$ and $73_{312}=3^7$.

I got both of these by solving your equation for the base $B$: $B=\frac{Y^x-Y}{X}$. You then need solutions where $B>X$. One thing to note is that if $Y$ is prime, then $B$ works out to be an integer because of Euler's Theorem.