Three-term arithmetic progressions of consecutive powerful numbers

Wouter van Doorn

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arXiv · 2605.06697

Three-term arithmetic progressions of consecutive powerful numbers

Wouter van Doorn

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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.

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Wouter van Doorn. 2026-05-04. Three-term arithmetic progressions of consecutive powerful numbers. https://arxiv.org/abs/2605.06697

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