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

0583475 - MÚ 2025 RIV BG eng J - Journal Article
Somer, L. - Křížek, Michal
Second-order linear recurrences with identically distributed residues modulo p^e.
Notes on Number Theory and Discrete Mathematics. Roč. 30, č. 1 (2024), s. 47-66. ISSN 1310-5132. E-ISSN 2367-8275
R&D Projects: GA ČR(CZ) GA24-10586S
Institutional support: RVO:67985840
Keywords : Lucas sequences * discriminant * second-order recurrence
OECD category: Pure mathematics
Method of publishing: Open access
https://doi.org/10.7546/nntdm.2024.30.1.47-66

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}.
Permanent Link: https://hdl.handle.net/11104/0351442