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

Abstract

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 $d\in\mathbb Z_p$, the frequency with which $d$ appears in a full period of $u(a,-1)\pmod p$ is the same frequency as $cd$ appears in $u(a',-1)\pmod p$. Here $u(a,b)$ satisfies the recursion relation $u_{n+2}=au_{n+1}+bu_n$ with initial terms $u_0=0$ and $u_1=1$. 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 $u(a,1)\pmod p$ given in a previous article. We further generalize our results by showing for a certain class of primes $p$ that if $e>1$, $b=\pm 1$, 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 $d\pmod{p^e}$, the frequency with which $d$ appears in $u(a,b)\pmod{p^e}$ is the same frequency as $cd$ appears in $u(a',b)\pmod{p^e}$.