Perfect Powers: Pillai's works and their developments

Overview & Synopsis

A perfect power is a positive integer of the form $a^x$ where $a\ge 1$ and $x\ge 2$ are rational integers. Subbayya Sivasankaranarayana Pillai wrote several papers on these numbers. In 1936 and again in 1945 he suggested that for any given $k\ge 1$, the number of positive integer solutions $(a,\, b,\, x,\, y)$, with $x\ge 2$ and $y\ge 2$, to the Diophantine equation $a^x-b^y=k$ is finite. This conjecture amounts to saying that the distance between two consecutive elements in the sequence of perfect powers tends to infinity. After a short introduction to Pillai's work on Diophantine questions, we quote some later developments and we discuss related open problems.