The Sum of Irreducible Fractions with Consecutive Denominators Is Never an Integer in PA^-

Pambuccian, Victor

Notre Dame J. Formal Logic, Tome 49 (2008) no. 1, p. 425-429 / Harvested from Project Euclid

Résumé

Two results of elementary number theory, going back to Kürschák and Nagell, stating that the sums $\sum_{i=1}^k \frac{m_i}{n+i}$ (with $k\geq 1$ , $(m_i, n+i)=1$ , $m_i\lessthan n+i$ ) and $\sum_{i=0}^k \frac{1}{m+in}$ (with $n, m, k$ positive integers) are never integers, are shown to hold in $\mathrm{PA}^{-}$ , a very weak arithmetic, whose axiom system has no induction axiom.

Détail

Publié le : 2008-10-15
Classification:  Kaye's $\mathrm{PA]^{-}$,  Kürschák's theorem,  Nagell's theorem,  weak arithmetic,  03C62,  03B30,  11A05