The sum of irreducible fractions with consecutive denominators is never an integer in a very weak arithmetic

2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/146494Two theorems of elmentary arithmetic, one stating that the sum of the reciprocals of any number of consecutive positive integers is never an integer, and a generalization thereof by Trygve Nagell, are shown to be provable inside a very weak arithmetic, Richard Kaye's $PA^-$, in which there is no induction axiom whatsoever.LogicNumber Theory03F30; 11A99The sum of irreducible fractions with consecutive denominators is never an integer in a very weak arithmetictext