The sum of irreducible fractions with consecutive denominators is never an integer in \(\mathrm{PA}^-\)
The sum \(1+1/2+\dots +1/n\) is never an integer. Pambuccian considers several generalizations of this statement and asks how elementary can their proofs be. The main result is that the following theorem is provable in the weak system \(\text{PA}^-\) which consists of the axioms for positive parts of discretely ordered rings in the language \(\{+,\times, <, 0, 1\}\) (and no induction axioms). The theorem states that if \(k\geq 1\), for all \(i\leq k\), \(m_i\) and \(n+i\) are relatively prime, and \(m_i<n+i\), then the sum \(m_0/n+m_1/(n+1)+\dots+ m_k/(n+k)\) is never an integer.
scientific article; zbMATH DE number 5657677