Journal of Integer Sequences, Vol. 26 (2023), Article 23.1.1

Arithmetic Progressions Among Powerful Numbers


Tsz Ho Chan
Mathematics Department
Kennesaw State University
Marietta, GA 30060
USA

Abstract:

In this paper, we study k-term arithmetic progressions N, N + d, ..., N + (k – 1)d of powerful numbers. Unconditionally, we exhibit infinitely many 3-term arithmetic progressions of powerful numbers with d ≤ 5N½. Assuming the abc-conjecture, we obtain a nearly tight lower bound on the common difference. We also prove some partial results when k ≥ 4 and pose some open questions.


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(Concerned with sequences A001694.)


Received September 30 2022; revised versions received October 1 2022; October 3 2022; December 13 2022. Published in Journal of Integer Sequences, December 14 2022.


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