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

TL;DR

This paper proves that certain sums of reciprocals of consecutive integers are never integers within a very weak arithmetic system, extending classical theorems without using induction.

Contribution

It demonstrates that two elementary arithmetic theorems can be proved in a weak arithmetic system lacking induction, highlighting their fundamental logical strength.

Findings

Sum of reciprocals of consecutive integers is never an integer.

Theorems are provable in Richard Kaye's $PA^-$ system without induction.

Extends classical results to a minimal logical framework.

Abstract

Two 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.