The sum of irreducible fractions with consecutive denominators is never an integer in \(\mathrm{PA}^-\)
type: Item
instance of: scholarly article
title: The sum of irreducible fractions with consecutive denominators is never an integer in \(\mathrm{PA}^-\)
zbMATH Open document ID: 1185.03086
DOI: 10.1215/00294527-2008-021
author: Victor V. Pambuccian
published in: Notre Dame Journal of Formal Logic
publication date: 2010-01-13T00:00:00Z
review text: The sum \(1+1/2+\dots +1/n\) is never an integer. Pambuccian considers several generalizations of this statement and asks how elementary can their proofs be. The main result is that the following theorem is provable in the weak system \(\text{PA}^-\) which consists of the axioms for positive parts of discretely ordered rings in the language \(\{+,\times, <, 0, 1\}\) (and no induction axioms). The theorem states that if \(k\geq 1\), for all \(i\leq k\), \(m_i\) and \(n+i\) are relatively prime, and \(m_i<n+i\), then the sum \(m_0/n+m_1/(n+1)+\dots+ m_k/(n+k)\) is never an integer.
reviewed by: Roman Kossak
Mathematics Subject Classification ID: 03F30
Mathematics Subject Classification ID: 03B30
Mathematics Subject Classification ID: 03C62
Mathematics Subject Classification ID: 11A05
zbMATH DE Number: 5657677
zbMATH Keywords: Kaye's \(\text{PA}^-\)
zbMATH Keywords: Kürschák's theorem
zbMATH Keywords: Nagell's theorem
zbMATH Keywords: weak arithmetic
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/00294527-2008-021
OpenAlex ID: W2054575177
description: scientific article; zbMATH DE number 5657677