On Eswarathasan--Levine and Boyd's conjectures for harmonic numbers
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| Format: | Preprint |
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2025
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| _version_ | 1866909545149759488 |
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| author | Carofiglio, Leonardo Cherubini, Giacomo Gambini, Alessandro |
| author_facet | Carofiglio, Leonardo Cherubini, Giacomo Gambini, Alessandro |
| contents | We provide numerical evidence towards three conjectures on harmonic numbers by Eswarathasan--Levine and Boyd. Let $J_p$ denote the set of integers $n\geq 1$ such that the harmonic number $H_n$ is divisible by a prime $p$. The conjectures state that: $(i)$ $J_p$ is always finite and of the order $O(p^2(\log\log p)^{2+ε})$; $(ii)$ the set of primes for which $J_p$ is minimal (called harmonic primes) has density $e^{-1}$ among all primes; $(iii)$ no harmonic number is divisible by $p^4$. We prove $(i)$ and $(iii)$ for all $p\leq 16843$ with at most one exception, and enumerate harmonic primes up to~$50\cdot 10^5$, finding a proportion close to the expected density. Our work extends previous computations by Boyd by a factor of about $30$ and $50$, respectively. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_15714 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Eswarathasan--Levine and Boyd's conjectures for harmonic numbers Carofiglio, Leonardo Cherubini, Giacomo Gambini, Alessandro Number Theory Primary 11B83, Secondary 11Y55, 11Y70 We provide numerical evidence towards three conjectures on harmonic numbers by Eswarathasan--Levine and Boyd. Let $J_p$ denote the set of integers $n\geq 1$ such that the harmonic number $H_n$ is divisible by a prime $p$. The conjectures state that: $(i)$ $J_p$ is always finite and of the order $O(p^2(\log\log p)^{2+ε})$; $(ii)$ the set of primes for which $J_p$ is minimal (called harmonic primes) has density $e^{-1}$ among all primes; $(iii)$ no harmonic number is divisible by $p^4$. We prove $(i)$ and $(iii)$ for all $p\leq 16843$ with at most one exception, and enumerate harmonic primes up to~$50\cdot 10^5$, finding a proportion close to the expected density. Our work extends previous computations by Boyd by a factor of about $30$ and $50$, respectively. |
| title | On Eswarathasan--Levine and Boyd's conjectures for harmonic numbers |
| topic | Number Theory Primary 11B83, Secondary 11Y55, 11Y70 |
| url | https://arxiv.org/abs/2503.15714 |