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

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

2007-08-17

arxiv.org/abs/0708.2439v1

Mathematics

Number Theory

17 pages, 2 tables

Scientific paper

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.

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