### Abstract

Two results of elementary number theory, going back to Kürschák and Nagell, stating that the sums Σ_{i}
^{k}=1 mi/n+i (with k ≥ 1, (m_{i},n + i) = 1, mi < n + i) and Σ_{i}
^{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 language | English (US) |
---|---|

Pages (from-to) | 425-429 |

Number of pages | 5 |

Journal | Notre Dame Journal of Formal Logic |

Volume | 49 |

Issue number | 4 |

DOIs | |

State | Published - 2008 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

}

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 -