The sum of irreducible fractions with consecutive denominators is never an integer in PA-

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Two results of elementary number theory, going back to Kürschák and Nagell, stating that the sums Σi k=1 mi/n+i (with k ≥ 1, (mi,n + i) = 1, mi < n + i) and Σi k=0 1/m+in (with n, m, k positive integers) are never integers, are shown to hold in PA-, a very weak arithmetic, whose axiom system has no induction axiom.

Original languageEnglish (US)
Pages (from-to)425-429
Number of pages5
JournalNotre Dame Journal of Formal Logic
Volume49
Issue number4
DOIs
StatePublished - 2008

Fingerprint

Irreducible fraction
Denominator
Axiom
Consecutive
Integer
Number theory
Proof by induction

Keywords

  • Kaye's PA
  • Kürschák's theorem
  • Nagell's theorem
  • Weak arithmetic

ASJC Scopus subject areas

  • Logic

Cite this

The sum of irreducible fractions with consecutive denominators is never an integer in PA- . / Pambuccian, Victor.

In: Notre Dame Journal of Formal Logic, Vol. 49, No. 4, 2008, p. 425-429.

Research output: Contribution to journalArticle

@article{9f52609a1b9641d9bba9efe667a3b43c,
title = "The sum of irreducible fractions with consecutive denominators is never an integer in PA-",
abstract = "Two results of elementary number theory, going back to K{\"u}rsch{\'a}k and Nagell, stating that the sums Σi k=1 mi/n+i (with k ≥ 1, (mi,n + i) = 1, mi < n + i) and Σi k=0 1/m+in (with n, m, k positive integers) are never integers, are shown to hold in PA-, a very weak arithmetic, whose axiom system has no induction axiom.",
keywords = "Kaye's PA, K{\"u}rsch{\'a}k's theorem, Nagell's theorem, Weak arithmetic",
author = "Victor Pambuccian",
year = "2008",
doi = "10.1215/00294527-2008-021",
language = "English (US)",
volume = "49",
pages = "425--429",
journal = "Notre Dame Journal of Formal Logic",
issn = "0029-4527",
publisher = "Duke University Press",
number = "4",

}

TY - JOUR

T1 - The sum of irreducible fractions with consecutive denominators is never an integer in PA-

AU - Pambuccian, Victor

PY - 2008

Y1 - 2008

N2 - Two results of elementary number theory, going back to Kürschák and Nagell, stating that the sums Σi k=1 mi/n+i (with k ≥ 1, (mi,n + i) = 1, mi < n + i) and Σi k=0 1/m+in (with n, m, k positive integers) are never integers, are shown to hold in PA-, a very weak arithmetic, whose axiom system has no induction axiom.

AB - Two results of elementary number theory, going back to Kürschák and Nagell, stating that the sums Σi k=1 mi/n+i (with k ≥ 1, (mi,n + i) = 1, mi < n + i) and Σi k=0 1/m+in (with n, m, k positive integers) are never integers, are shown to hold in PA-, a very weak arithmetic, whose axiom system has no induction axiom.

KW - Kaye's PA

KW - Kürschák's theorem

KW - Nagell's theorem

KW - Weak arithmetic

UR - http://www.scopus.com/inward/record.url?scp=84875514235&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84875514235&partnerID=8YFLogxK

U2 - 10.1215/00294527-2008-021

DO - 10.1215/00294527-2008-021

M3 - Article

AN - SCOPUS:84875514235

VL - 49

SP - 425

EP - 429

JO - Notre Dame Journal of Formal Logic

JF - Notre Dame Journal of Formal Logic

SN - 0029-4527

IS - 4

ER -