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

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F24%3A00583475" target="_blank" >RIV/67985840:_____/24:00583475 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.7546/nntdm.2024.30.1.47-66" target="_blank" >https://doi.org/10.7546/nntdm.2024.30.1.47-66</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.7546/nntdm.2024.30.1.47-66" target="_blank" >10.7546/nntdm.2024.30.1.47-66</a>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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 dinmathbb 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 dpmod{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}.

Název v anglickém jazyce

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

Popis výsledku anglicky

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 dinmathbb 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 dpmod{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}.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA24-10586S" target="_blank" >GA24-10586S: Analytické a numerické modelování hysterezních jevů</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2024

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Notes on Number Theory and Discrete Mathematics

ISSN

1310-5132

e-ISSN

2367-8275

Svazek periodika

30

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

BG - Bulharská republika

Počet stran výsledku

20

Strana od-do

47-66

Kód UT WoS článku

001221794100003

EID výsledku v databázi Scopus

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