An extension of Boyd's $p$-adic algorithm for the harmonic series

TL;DR

This paper extends Boyd's p-adic algorithm to analyze the size of specific sets related to harmonic series modulo primes, providing extensive computational results and exploring potential applications to Fibonacci sums.

Contribution

The paper introduces an extended p-adic algorithm for studying harmonic series modulo primes, with comprehensive computational analysis for primes up to 100 and initial results on Fibonacci-related sums.

Findings

Most sets J_p(y) are finite for p among the first 100 odd primes.

For p=13 and y=9, the set J_p(y) has size 18763.

The method has potential applications to sums involving Fibonacci numbers.

Abstract

In this paper we will extend a $p$-adic algorithm of Boyd in order to study the size of the set: \[J_p(y)=\left\{n :\sum_{j=1}^{n}\frac{y^j}{j}\equiv 0(\mod p)\right\}.\] Suppose that $p$ is one of the first 100 odd primes and $y\in\{1,2,...,p-1\}$, then our calculations prove that $|J_p(y)|<\infty$ in 24240 out of 24578 possible cases. Among other results we show that $|J_{13}(9)|=18763$. The paper concludes by discussing some possible applications of our method to sums involving Fibonacci numbers.