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arXiv · 2303.15010

$p$-adic valuation of harmonic sums and their connections with Wolstenholme primes

Leonardo Carofiglio·Luigi De Filpo·Alessandro Gambini
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Abstract

We explore a conjecture posed by Eswarathasan and Levine on the distribution of $p$-adic valuations of harmonic numbers $H(n)=1+1/2+\cdots+1/n$ that states that the set $J_p$ of the positive integers $n$ such that $p$ divides the numerator of $H(n)$ is finite. We proved two results, using a modular-arithmetic approach, one for non-Wolstenholme primes and the other for Wolstenholme primes, on an anomalous asymptotic behaviour of the $p$-adic valuation of $H(p^mn)$ when the $p$-adic valuation of $H(n)$ equals exactly 3.

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Leonardo Carofiglio, Luigi De Filpo, Alessandro Gambini. 2023-03-27. $p$-adic valuation of harmonic sums and their connections with Wolstenholme primes. https://doi.org/10.1007/s13226-023-00387-1

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