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

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.