Easy criteria to determine if a prime divides certain second-order recurrences

Result code in IS VaVaI

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Result on the web

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DOI - Digital Object Identifier

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Result language

angličtina

Original language name

Easy criteria to determine if a prime divides certain second-order recurrences

Original language description

Let F(a, b) denote the set of all second-order recurrences w(a, b) satisfying the recursion relation wn+2 = awn+1 + bwn, where the discriminant D = a2+4b and a, b,w0, and w1 are all integers. Let u(a, b) denote the recurrence with initial terms u0 = 0 and u1 = 1. We say that the prime p is a divisor of w(a, b) if p | wn for some integer n >= 0. Let z(p) denote the least positive integer n such that un = 0 (mod p). Then z(p) | p (D/p), where (D/p) denotes the Legendre symbol. Define the index i(p) as i(p) = p (D/p) z(p). When i(p) = 1 or 2, we will find easy criteria to determine exactly when p is a divisor of w(a, b) based on the residue class or quadratic character of w2 1 aw1w0 bw2 0 modulo p. This generalizes results of Vandervelde when a = b = 1.

Czech name

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Czech description

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Type

J_x - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)

CEP classification

BA - General mathematics

OECD FORD branch

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Project

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Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2013

Confidentiality

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

Name of the periodical

Fibonacci Quarterly

ISSN

0015-0517

e-ISSN

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Volume of the periodical

51

Issue of the periodical within the volume

1

Country of publishing house

US - UNITED STATES

Number of pages

10

Pages from-to

3-12

UT code for WoS article

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EID of the result in the Scopus database

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