Perfect powers in sequences of polygonal numbers

Abstract

Let $P_s(n)$ denote the $n$-th $s$-gonal number. Consider the Diophantine equation $P_s(n)=t^m$ for integers $n,s,t$ and $m>2$. All solutions to this equation are known for $m>2$ and $s\in \{3,5,6,8,10,20\}$. Here we extend these results to the cases $s=2k+4$ (where $k=4,6$ or $5\le k\le 97$ is a prime number) and $s=k+4$ (where $k=9,15$ or $3\le k\le 97$ is a prime number). The proofs of our results use the modular and hypergeometric methods, linear forms in logarithms and extensive calculations. We were unable to completely solve the above Diophantine equations, but we expect (based on GRH and the weak effective $abc$ conjecture) that there will be no additional solutions beyond those explicitly shown in Theorems~1, 2 and 3.

published version, but comments are still welcome