# A certain reciprocal power sum is never an integer

## TL;DR

This paper proves that certain reciprocal power sums generated by polynomials of degree at least two are never integers and, under specific conditions, can be arbitrarily close to 1, revealing new properties of these sums.

## Contribution

It establishes that these sums cannot equal 1 and demonstrates their density near 1 for quadratic polynomials, using analytic and p-adic methods along with a result of Kakeya.

## Key findings

- Reciprocal power sums are never integers for degree ≥ 2 polynomials.
- These sums can be arbitrarily close to 1, showing density in a specific interval.
- The results extend previous work on linear polynomials to higher degrees.

## Abstract

By $(\mathbb{Z}^+)^{\infty}$ we denote the set of all the infinite sequences $\mathcal{S}=\{s_i\}_{i=1}^{\infty}$ of positive integers (note that all the $s_i$ are not necessarily distinct and not necessarily monotonic). Let $f(x)$ be a polynomial of nonnegative integer coefficients. Let $\mathcal{S}_n:=\{s_1, ..., s_n\}$ and $H_f(\mathcal{S}_n):=\sum_{k=1}^{n}\frac{1}{f(k)^{s_{k}}}$. When $f(x)$ is linear, Feng, Hong, Jiang and Yin proved in [A generalization of a theorem of Nagell, Acta Math. Hungari, in press] that for any infinite sequence $\mathcal{S}$ of positive integers, $H_f(\mathcal{S}_n)$ is never an integer if $n\ge 2$. Now let deg$f(x)\ge 2$. Clearly, $0<H_f(\mathcal{S}_n)<\zeta(2)<2$. But it is not clear whether the reciprocal power sum $H_f(\mathcal{S}_n)$ can take 1 as its value. In this paper, with the help of a result of Erd\H{o}s, we use the analytic and $p$-adic method to show that for any infinite sequence $\mathcal{S}$ of positive integers and any positive integer $n\ge 2$, $H_f(\mathcal{S}_n)$ is never equal to 1. Furthermore, we use a result of Kakeya to show that if $\frac{1}{f(k)}\le\sum_{i=1}^\infty\frac{1}{f(k+i)}$ holds for all positive integers $k$, then the union set $\bigcup\limits_{\mathcal{S}\in (\mathbb{Z}^+)^{\infty}} \{ H_f(\mathcal{S}_n) | n\in \mathbb{Z}^+ \}$ is dense in the interval $(0,\alpha_f)$ with $\alpha_f:=\sum_{k=1}^{\infty}\frac{1}{f(k)}$. It is well known that $\alpha_f= \frac{1}{2}\big(\pi \frac{e^{2\pi}+1}{e^{2\pi}-1}-1\big)\approx 1.076674$ when $f(x)=x^2+1$. Our dense result infers that when $f(x)=x^2+1$, for any sufficiently small $\varepsilon >0$, there are positive integers $n_1$ and $n_2$ and infinite sequences $\mathcal{S}^{(1)}$ and $\mathcal{S}^{(2)}$ of positive integers such that $1-\varepsilon<H_f(\mathcal{S}^{(1)}_{n_1})<1$ and $1<H_f(\mathcal{S}^{(2)}_{n_2})<1+\varepsilon$.