On harmonic numbers and Lucas sequences

TL;DR

This paper explores new congruences involving harmonic numbers and Lucas sequences, establishing results that connect these mathematical concepts through novel theorems and modular identities.

Contribution

It introduces the first systematic study of congruences linking harmonic numbers with Lucas sequences, including a key theorem on sums modulo primes.

Findings

Proves a congruence involving harmonic numbers and Lucas sequences for primes p>5.

Identifies conditions on p for the sum to vanish modulo p.

Establishes a new connection between harmonic numbers and Lucas sequences in number theory.

Abstract

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.