A Unique Perfect Power Decagonal Number

Abstract

Let ${\mathcal{P}}_s(n)$ denote the $n$th $s$-gonal number. We consider the equation \[\mathcal{P}_s(n) = y^m, \] for integers $n,s,y,$ and $m$. All solutions to this equation are known for $m>2$ and $s\in \{3,5,6,8,20\}$. We consider the case $s=10$, that of decagonal numbers. Using a descent argument and the modular method, we prove that the only decagonal number $>1$ expressible as a perfect $m$th power with $m>1$ is ${\mathcal{P}}_{10}(3)=3^3$.

5 pages. Some typos corrected

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