On the unsolvability of certain equations of Erd\H{o}s-Moser type
| Autor: | Ioulia N. Baoulina |
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| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: | |
| Popis: | 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 $S_k(m+1)/S_k(m)$ is never an integer, or, equivalently, that for any positive integer $a$, the equation $aS_k(m)=m^k$ has no solutions in positive integers $k$ and $m$ with $m>3$. In this paper, we show that for many integers $a$ the equation $aT_k(m)=(2m+1)^k$, 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 $T_k(m+1)/T_k(m)$ is never an integer. Comment: 8 pages |
| Databáze: | OpenAIRE |
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