On harmonic numbers and Lucas sequences (Q2915552)

Let \(H_n=1+1/2+\cdots+1/n\) be the \(n\)th harmonic number. Inspired by Wolstenholme's congruence \(H_{p-1}\equiv 0\pmod {p^2}\) valid for primes \(p>3\), in prior work this author has proved several congruences including sums of squares and cubes of harmonic numbers modulo primes. In the paper under review, he studies congruences involving harmonic numbers and members of the Lucas sequence \(\{u_n\}_{n\geq 0}\) defined by \(u_0=0\), \(u_1=1\) and \(u_{n+2}=Au_{n+1}-Bu_n\) for \(n\geq 0\). Specializing to particular values of \(A\) and \(B\), he obtains interesting congruences involving harmonic numbers and Fibonacci numbers such as \(\sum_{k=0}^{p-1} F_k H_k^n\equiv 0\pmod p\) for all \(n\geq 0\) and all primes \(p\equiv \pm 1\pmod 5\). The author also conjectures a total of 13 congruences involving harmonic numbers, members of certain Lucas sequences and harmonic numbers of the second kind defined as \(H_n^{(2)}=1+1/2^2+ \cdots +1/n^2\), which should spur further research in the area.