Second-order linear recurrences with identically distributed residues modulo pᵉ

Let p be an odd prime and let u (a, -1) and u (a', -1) be two Lucas sequences whose discriminants have the same nonzero quadratic character modulo p and whose periods modulo p are equal. We prove that there is then an integer c such that for all d Zₚ, the frequency with which d appears in a full period of u (a, -1) p is the same frequency as cd appears in u (a', -1) p. Here u (a, b) satisfies the recursion relation u₍+₂=au₍+₁+buₙ with initial terms u₀=0 and u₁=1. Similar results are obtained for the companion Lucas sequences v (a, -1) and v (a', -1). This paper extends analogous statements for Lucas sequences of the form u (a, 1) p given in a previous article. We further generalize our results by showing for a certain class of primes p that if e>1, b= 1, and u (a, b) and u (a', b) are Lucas sequences with the same period modulo p, then there exists an integer c such that for all residues dpᵉ, the frequency with which d appears in u (a, b) pᵉ is the same frequency as cd appears in u (a', b) pᵉ.