Harmonic prime

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title: "Harmonic prime" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["classes-of-prime-numbers"] topic_path: "general/classes-of-prime-numbers" source: "https://en.wikipedia.org/wiki/Harmonic_prime" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

A harmonic prime is a prime number that divides the numerators of exactly three harmonic numbers.

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

For example, the numerators of the fractions given by \sum_{i=1}^{4} \frac{1}{i} , \sum_{i=1}^{20} \frac{1}{i} , and \sum_{i=1}^{24} \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.

References

References

  1. (1994). "A ''p''-adic Study of the Partial Sums of the Harmonic Series". [[Experimental Mathematics (journal).

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