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.
p-adic valuation of harmonic sums and their connections with Wolstenholme primes / Carofiglio, Leonardo; De Filpo, Luigi; Gambini, Alessandro. - In: INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS. - ISSN 0019-5588. - (2023), pp. 1-12. [10.1007/s13226-023-00387-1]
p-adic valuation of harmonic sums and their connections with Wolstenholme primes
Carofiglio, Leonardo;De Filpo, Luigi;Gambini, Alessandro
2023
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.File | Dimensione | Formato | |
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