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Second-order linear recurrences with identically distributed residues modulo p^e
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SYSNO ASEP 0583475 Druh ASEP J - Článek v odborném periodiku Zařazení RIV J - Článek v odborném periodiku Poddruh J Článek ve WOS Název Second-order linear recurrences with identically distributed residues modulo p^e Tvůrce(i) Somer, L. (US)
Křížek, Michal (MU-W) RID, SAI, ORCIDZdroj.dok. Notes on Number Theory and Discrete Mathematics - ISSN 1310-5132
Roč. 30, č. 1 (2024), s. 47-66Poč.str. 20 s. Forma vydání Tištěná - P Jazyk dok. eng - angličtina Země vyd. BG - Bulharsko Klíč. slova Lucas sequences ; discriminant ; second-order recurrence Vědní obor RIV BA - Obecná matematika Obor OECD Pure mathematics CEP GA24-10586S GA ČR - Grantová agentura ČR Způsob publikování Open access Institucionální podpora MU-W - RVO:67985840 UT WOS 001221794100003 DOI 10.7546/nntdm.2024.30.1.47-66 Anotace 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}. Pracoviště Matematický ústav Kontakt Jarmila Štruncová, struncova@math.cas.cz, library@math.cas.cz, Tel.: 222 090 757 Rok sběru 2025 Elektronická adresa https://doi.org/10.7546/nntdm.2024.30.1.47-66
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