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\vskip 1cm{\LARGE\bf  On the $p$-Adic Valuations of \\
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Sums of Powers of Integers}
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\large
Gombodorj Bayarmagnai and Sainjargal Delger \\  
Department of Mathematics\\
National University of Mongolia\\
Baga Toirog \\
Ulaanbaatar 14200\\
Mongolia\\
\href{mailto:bayarmagnai@num.edu.mn}{\tt bayarmagnai@num.edu.mn}\\
\href{mailto:sdelger96@gmail.com}{\tt sdelger96@gmail.com} \\
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\begin{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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