A generalization of a theorem of Nagell

Abstract

Let n be a positive integer. Theisinger [7] proved that if ${n\ge 2}$ , then the n-th harmonic sum ${\sum_{k=1}^n\frac{1}{k}}$ is not an integer. Let a and b be positive integers. Nagell [6] extended Theisinger’s theorem by showing that the reciprocal sum ${\sum_{k=1}^{n}\frac{1}{a+(k-1)b}}$ is not an integer if ${n\ge 2}$ . Erdős and Niven [2] proved a theorem of a similar nature that states that there is only a finite number of integers n for which one or more of the elementary symmetric functions of ${1, 1/2, \ldots, 1/n}$ is an integer. We present a generalization of Nagell’s theorem. In fact, we show that for arbitrary n positive integers ${s_1, \ldots, s_n}$ (not necessarily distinct and not necessarily monotonic), the reciprocal power sum

$$\sum_{k=1}^{n}\frac{1}{(a+(k-1)b)^{s_{k}}}$$

is never an integer if ${n\ge 2}$ . The proof of our result is analytic and p-adic in character.