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

Author

Listed:
  • Leonardo Carofiglio

    (Sapienza University of Rome)

  • Luigi Filpo

    (Sapienza University of Rome)

  • Alessandro Gambini

    (Sapienza University of Rome)

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.

Suggested Citation

  • Leonardo Carofiglio & Luigi Filpo & Alessandro Gambini, 2024. "p-adic valuation of harmonic sums and their connections with Wolstenholme primes," Indian Journal of Pure and Applied Mathematics, Springer, vol. 55(2), pages 555-566, June.
    • Handle: RePEc:spr:indpam:v:55:y:2024:i:2:d:10.1007_s13226-023-00387-1
    DOI: 10.1007/s13226-023-00387-1
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