Differences of Harmonic Numbers and the $abc$-Conjecture

TL;DR

This paper explores which numbers can be expressed as differences of harmonic numbers, examines their connection to the $abc$-conjecture, and identifies special classes of such numbers including primes and famous prime types.

Contribution

It characterizes all numbers less than 100 that cannot be written as differences of harmonic numbers and extends the $abc$-conjecture's validity to new sets including these numbers and special primes.

Findings

Only eleven numbers less than 100 are not differences of harmonic numbers.

There are infinitely many non-difference harmonic numbers, including primes of the form 48k+41.

Certain Fermat and Mersenne primes are either differences of harmonic numbers or have known representations.

Abstract

Our main source of inspiration was a talk by Hendrik Lenstra on harmonic numbers, which are numbers whose only prime factors are two or three. Gersonides proved 675 years ago that one can be written as a difference of harmonic numbers in only four ways: 2-1, 3-2, 4-3, and 9-8. We investigate which numbers other than one can or cannot be written as a difference of harmonic numbers and we look at their connection to the $abc$-conjecture. We find that there are only eleven numbers less than 100 that cannot be written as a difference of harmonic numbers (we call these $ndh$-numbers). The smallest $ndh$-number is 41, which is also Euler's largest lucky number and is a very interesting number. We then show there are infinitely many $ndh$-numbers, some of which are the primes congruent to $41$ modulo $48$. For each Fermat or Mersenne prime we either prove that it is an $ndh$-number or find all ways it can be written as a difference of harmonic numbers. Finally, as suggested by Lenstra in his talk, we interpret Gersonides' theorem as "The $abc$-conjecture is true on the set of harmonic numbers" and we expand the set on which the $abc$-conjecture is true by adding to the set of harmonic numbers the following sets (one at a time): a finite set of $ndh$-numbers, the infinite set of primes of the form $48k+41$, the set of Fermat primes, and the set of Mersenne primes.