We provide numerical evidence towards three conjectures on harmonic numbers by Eswarathasan, Levine and Boyd. Let Jp denote the set of integers n ≤1 such that the harmonic number Hn is divisible by a prime p. The conjectures state that: (i) Jp is always finite and of the order O(p2(log log p) 2+); (ii) the set of primes for which Jp is minimal (called harmonic primes) has density e-1 among all primes; (iii) no harmonic number is divisible by p4. We prove parts (i) and (iii) for all p ≤ 16843 with at most one exception, and enumerate harmonic primes up to 50 105, finding a proportion close to the expected density. Our work extends previous computations by Boyd by a factor of approximately 30 and 50, respectively.
ON ESWARATHASAN–LEVINE AND BOYD’S CONJECTURES FOR HARMONIC NUMBERS
LEONARDO CAROFIGLIO;
2025-01-01
Abstract
We provide numerical evidence towards three conjectures on harmonic numbers by Eswarathasan, Levine and Boyd. Let Jp denote the set of integers n ≤1 such that the harmonic number Hn is divisible by a prime p. The conjectures state that: (i) Jp is always finite and of the order O(p2(log log p) 2+); (ii) the set of primes for which Jp is minimal (called harmonic primes) has density e-1 among all primes; (iii) no harmonic number is divisible by p4. We prove parts (i) and (iii) for all p ≤ 16843 with at most one exception, and enumerate harmonic primes up to 50 105, finding a proportion close to the expected density. Our work extends previous computations by Boyd by a factor of approximately 30 and 50, respectively.
| File | Dimensione | Formato | |
|---|---|---|---|
|
on-eswarathasan-levine-and-boyds-conjectures-for-harmonic-numbers.pdf
Accesso aperto
Descrizione: PDF articolo
Tipo di file:
PDF EDITORIALE
Dimensione
399.06 kB
Formato
Adobe PDF
|
399.06 kB | Adobe PDF | Visualizza/Apri |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
