Differences of Harmonic Numbers and the abc-Conjecture

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's 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.