Second-order linear recurrences with identically distributed residues modulo p^e

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 uₙ₊₂=auₙ₊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.