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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
bayarmagnai@num.edu.mn
sdelger96@gmail.com

in

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.