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, Erd\H{ o}s 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 $$\sum\limits_{ k=1}^{ n}\frac{ 1}{ (a+(k-1)b)^{ s_{ k}}}$$ is never an integer if $n\ge 2$. The proof of our result is analytic and $p$-adic in character.
Categories: math.NT