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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.15010 |
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| _version_ | 1866909230752071680 |
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| author | Carofiglio, Leonardo De Filpo, Luigi Gambini, Alessandro |
| author_facet | Carofiglio, Leonardo De Filpo, Luigi Gambini, Alessandro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_15010 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $p$-adic valuation of harmonic sums and their connections with Wolstenholme primes Carofiglio, Leonardo De Filpo, Luigi Gambini, Alessandro Number Theory 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. |
| title | $p$-adic valuation of harmonic sums and their connections with Wolstenholme primes |
| topic | Number Theory |
| url | https://arxiv.org/abs/2303.15010 |