Arithmetic progressions among powerful numbers

TL;DR

This paper investigates the existence and properties of arithmetic progressions within powerful numbers, establishing bounds under the abc-conjecture and demonstrating infinite progressions for three-term cases unconditionally.

Contribution

It provides new bounds on common differences of arithmetic progressions of powerful numbers under the abc-conjecture and proves the existence of infinitely many 3-term progressions unconditionally.

Findings

Under abc-conjecture, d  N^{1/2 - \u03b5}

Existence of infinitely many 3-term progressions with d  N^{1/2} unconditionally

Partial results and open questions for k  4 or more terms

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.