Brute-force search for solution to an unsolved mathematical inequality
A first step in optimizing would be to avoid computing the same values multiple times.

Secondly, I would drop the floating point arithmetic altogether. You have integer bases with integer exponents, so use integers. \$(\frac{3}{2})^k\$ = \$\frac{3^k}{2^k}\$ and applying floor also results in an integer.

I'm not familiar with the problem at hand, but the mention that none are known leads me to think that the values might be quite higher than 1000, otherwise a solution would have been trivial to find.