  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.
