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arXiv · 2210.00281

Arithmetic progressions among powerful numbers

Tsz Ho Chan
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Abstract

In this paper, we study $k$-term arithmetic progressions $N, N+d, ..., N+(k-1)d$ of powerful numbers. Under the $abc$-conjecture, we obtain $d \gg_\epsilon N^{1/2 - \epsilon}$. On the other hand, there exist infinitely many $3$-term arithmetic progressions of powerful numbers with $d \ll N^{1/2}$ unconditionally. We also prove some partial results when $k \ge 4$ and pose some open questions.

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Tsz Ho Chan. 2022-10-01. Arithmetic progressions among powerful numbers. https://arxiv.org/abs/2210.00281

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