Three-term arithmetic progressions of consecutive powerful numbers

TL;DR

The paper proves the existence of infinitely many three-term arithmetic progressions of powerful numbers with a specific common difference and conjectures about their consecutive nature, addressing a question posed by Erdős.

Contribution

It establishes the existence of infinitely many such progressions and proposes a conjecture about their consecutive terms, advancing understanding of powerful numbers.

Findings

Infinitely many three-term progressions with difference d=2√N+1 exist.

Conjecture that infinitely many progressions are of consecutive powerful numbers.

Addresses a question posed by Erdős regarding powerful numbers.

Abstract

We show that infinitely many three-term arithmetic progressions $N, N+d, N+2d$ of powerful numbers exist with $d = 2\sqrt{N} + 1$. We further conjecture that infinitely many of these progressions consist of three consecutive terms in the sequence of powerful numbers, which would answer a question of Erd\H{o}s in the negative.