A generalization of a theorem of Nagell

Abstract

Let $n$ be a positive integer. In 1915, Theisinger 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. In 1923, Nagell 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 $n\ge 2$. In 1946, Erdős and Niven 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 $1,1/2,...,1/n$ is an integer. In this paper, we present a generalization of Nagell's theorem. In fact, we show that for arbitrary $n$ positive integers $s_1,...,s_n$ (not necessarily distinct and not necessarily monotonic), the following 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.

12 pages. To appear in Acta Math. Hungar