harmonic prime

{{One source|date=January 2025}}

A harmonic prime {{OEIS|id=A092101}} is a prime number that divides the numerators of exactly three harmonic numbers.

Specifically, a harmonic prime {{mvar|p}} is always a factor of the numerators of the partial harmonic sums at positions {{math|p − 1}}, {{math|p² − p}}, and {{math|p² − 1}}.

For example, the numerators of the fractions given by $\backslash sum\_\{i=1\}^\{4\}\; \backslash frac\{1\}\{i\}$, $\backslash sum\_\{i=1\}^\{20\}\; \backslash frac\{1\}\{i\}$, and $\backslash sum\_\{i=1\}^\{24\}\; \backslash frac\{1\}\{i\}$ are 25, 55835135, and 1347822955, each of which is divisible by 5.

All prime numbers greater than 5 can also be found at those three indices, but many also appear at other indices. It is conjectured that there are infinitely many harmonic primes. {{Cite journal | last1 = Boyd | first1 = D. W. | title = A p-adic Study of the Partial Sums of the Harmonic Series | url = http://projecteuclid.org/euclid.em/1048515811 | doi = 10.1080/10586458.1994.10504298 | journal = Experimental Mathematics | volume = 3 | issue = 4 | pages = 287–302 | year = 1994 | id = CiteSeerX: {{URL|1=citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.56.7026|2=10.1.1.56.7026}} | zbl = 0838.11015 | url-status = live | archive-url = https://web.archive.org/web/20160127080246/http://projecteuclid.org/euclid.em/1048515811 | archive-date = 27 January 2016}}