A Generalization of E. Lehmer’s Congruence and Its Applications

Abstract

In this paper we announce the result that for any odd n > 1, $\sum\limits_{\scriptstyle i = 1 \hfill \atop \scriptstyle (i,n) = 1 \hfill} ^{\left( {n - 1} \right)/2} {\frac{1}{i} \equiv - 2{q_2}(n) + nq_2^2} (n)\quad (\bmod \;{n^2}),$ where ${q_r}(n) = ({r^{\phi (n)}} - 1)/n,{\text{ (r, n) = 1}}$ is Euler’s quotient of n with base r, which is a generalization of E. Lehmer’s congruence. As applications, we mention some generalizations of Morley’s congruence and Jacobstahl’s Theorem to modulo arbitary positive integers. The details of the proof will partly appear in Acta Arithmetica.