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

Abstract

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 $k\ge 1$, $(m_i,n+i)=1$, $m_i<n+i$) and $\sum _{i=0}^k\frac{1}{m+\mathrm{in}}$ (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.