On Eswarathasan--Levine and Boyd's conjectures for harmonic numbers
Leonardo Carofiglio, Giacomo Cherubini, Alessandro Gambini
TL;DR
The paper probes three conjectures of Eswarathasan--Levine and Boyd about harmonic numbers: the finiteness and size of $J_p$ for primes $p$, the density of harmonic primes, and the absence of $p$-adic divisibility beyond the third power. It advances the state of knowledge by performing large-scale computations up to $p\le 16843$ (with at most one exception) and enumerating harmonic primes up to $50\cdot 10^5$, using a $p$-adic, block-based recursion that exploits $H_{pn+k} \equiv H_k + \frac{H_n}{p} \pmod{p^2}$ to propagate information efficiently. The results confirm finiteness (except possibly $p=1381$), the predicted density $e^{-1}$ of harmonic primes, and the nonexistence of $\nu_p(H_n) \ge 4$ for the primes tested, while also detailing extinction times and valuations, thereby providing substantial numerical support for the conjectures and extending Boyd’s work by substantial factors. Overall, the study strengthens the probabilistic model's validity for harmonic primes and supplies extensive data to guide future theoretical work on $p$-adic divisibility patterns of harmonic numbers.
Abstract
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.