Are two integers equal in SOME base?

You can make the problem simpler by applying a little bit of number of theory. Taking your first example 13x7=28 we can expand this out into an explicit polynomial over the base: (1n+3)*7=2n+8.

If this solution has real roots then the roots are values of n (bases) for which the equation is true. If you like this sort of problem then you should read Computational Number Theory by Shoup. It's a fun book.