On harmonic numbers and Lucas sequences

Zhi-Wei Sun

Published 2010-01-04, updated 2011-05-23Version 2

Harmonic numbers $H_k=\sum_{0<j\le k}1/j (k=0,1,2,...)$ arise naturally in many fields of mathematics. In this paper we initiate the study of congruences involving both harmonic numbers and Lucas sequences. One of our three theorems is as follows: Let u_0=0, u_1=1, and u_{n+1}=u_n-4u_{n-1} for n=1,2,3,.... Then, for any prime p>5 we have $$\sum_{k=0}^{p-1}u_{k+\delta}H_k/2^k=0 (mod p),$$ where $\delta=0$ if p=1,2,4,8 (mod 15), and $\delta=1$ otherwise.