Journal of Integer Sequences, Vol. 25 (2022), Article 22.8.1

On the p-adic Valuations of Sums of Powers of Integers


Gombodorj Bayarmagnai and Sainjargal Delger
Department of Mathematics
National University of Mongolia
Baga Toirog
Ulaanbaatar 14200
Mongolia

Abstract:

In this paper we obtain a simple formula for the number of matching $p$-ary digits of certain terms of Lucas sequences for any odd prime $p$. Using this formula, we present a simple sufficient condition for the sequence $( v_p(a_1^n + a_2^n + \cdots + a_k^n) )_{n\geq 0}$ to be unbounded, where $a_1, a_2, \ldots, a_k$ ($k \geq 2$) are given integers and $v_p$ is the $p$-adic valuation.

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Received July 11 2022; revised versions received August 5 2022; August 31 2022; September 13 2022. Published in Journal of Integer Sequences, October 7 2022.

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