Generalizations of Pollock’s centered nonagonal numbers conjecture

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发表日期：

2025年11月3日

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Anji Dong,
Alexandru Zaharescu

摘要

We generalize a problem of Sir Frederick Pollock that is more than 170 years old. Among other things, we show that for each integer $k\geqslant 3$ $k ⩾ 3$ , any positive integer is a sum of at most $k+2$ $k + 2$ centered k-gonal numbers. We also show that for each prime $p\geqslant 7$ $p ⩾ 7$ , any large enough positive integer can be written as a sum of at most two p-gonal numbers plus at most three centered p-gonal numbers.