jmc

First, let us find which bases result in $555_{10}$ having four digits. We must find base b such that $b^4>555_{10}\ge b^3$. We easily determine that b can range from 5 to 8, inclusive. We can now try each of these four bases to see which yields a number in the form ABAB. For base six, we find that $6^3=216$, which can go into 555 only two times at most, leaving $555-2\cdot 216=123$ for the next three digits. $6^2=36$ goes into 123 three times at most, leaving us with $123-3\cdot 36=15$. Then, $6^1=6$ goes into 15 two times at most, leaving $15-2\cdot 6=3$ for the ones digit. So, the base $\boxed{6}$ equivalent of $555_{10}$ is $2323_6$, which satisfies all of the requirements stated.