On the unsolvability of certain equations of Erd\H{o}s-Moser type

Let $S_k(m):={\sum }_{j=1}^{m-1}{j}^k$ denote a power sum. In 2011, Kellner proposed the conjecture that for $m>3$ the ratio $S_k(m+1)\mathrm{/}{S}_k(m)$ is never an integer, or, equivalently, that for any positive integer $a$, the equation $a{S}_k(m)=m^k$ has no solutions in positive integers $k$ and $m$ with $m>3$. In this paper, we show that for many integers $a$ the equation $a{T}_k(m)=(2m+1{)}^k$, where $T_k(m):={\sum }_{j=1}^m(2j-1{)}^k$, has no solutions in positive integers $k$ and $m$. This leads us to the conjecture that for $m>1$ the ratio $T_k(m+1)\mathrm{/}{T}_k(m)$ is never an integer.