Arithmetic progressions among powerful numbers

2022/10/01 by Chan, Tsz Ho
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2210.00281

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 ≫_εN^{1/2 - ε}. On the other hand, there exist infinitely many 3-term arithmetic progressions of powerful numbers with d ≪ N^1/2 unconditionally. We also prove some partial results when k ≥ 4 and pose some open questions.

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