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

Leonardo Carofiglio (), Luigi Filpo () and Alessandro Gambini ()
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Leonardo Carofiglio: Sapienza University of Rome
Luigi Filpo: Sapienza University of Rome
Alessandro Gambini: Sapienza University of Rome

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 2, 555-566

Abstract: Abstract We explore a conjecture posed by Eswarathasan and Levine on the distribution of p-adic valuations of harmonic numbers $$H(n)=1+1/2+\cdots +1/n$$ H ( n ) = 1 + 1 / 2 + ⋯ + 1 / n that states that the set $$J_p$$ 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 $$H(p^mn)$$ H ( p m n ) when the p-adic valuation of H(n) equals exactly 3.

Keywords: Harmonic numbers; Harmonic sums; Wolstenholme primes (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-023-00387-1

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