On Lucas numbers, Lucas pseudoprimes and a numbertheoretical series involving Lucas pseudoprimes and Carmichael numbers

Andrzej Rotkiewicz

Published: 2007-09-28

Vol. 21 (2007)

On Lucas numbers, Lucas pseudoprimes and a numbertheoretical series involving Lucas pseudoprimes and Carmichael numbers

Andrzej Rotkiewicz

Abstract

The following theorems are proved:
(1) If α and β≠α are roots of the polynomial x²−Px+Q, where gcd(P,Q)=1, P = α+β is an odd positive integer, then (α+β)ⁿ⁺¹|α^x + β^x if and only if x = (2l+1)(α+β)ⁿ, where l = 0, 1, 2, . . . and then
gcd((α^{(α+β)^n}+β^{(α+β)^n})/(α+β)ⁿ⁺¹, α+β) = 1.
(2) Given integers P,Q with D = P²−4Q ≠ 0,−Q,−2Q,−3Q and ɛ=±1,
every arithmetic progression ax+b, where gcd(a,b)=1 contains an odd integer n₀ such that (D|n₀)=ɛ. The series Σ_n=1^∞1/logP_n^(a)=1, where P_n^(a)is the n-th strong Lucas pseudoprime with parameters P and Q of the form ax+b, where gcd(a,b)=1 such that (D|P_n^(a))=ɛ, is divergent.
(3) Let C_n denote the n-th Carmichael number. From the conjecture of P. Erdős that C(x) > x^1−ɛ for every ɛ>0 and x ≥ x₀(ɛ), where C(x) denotes the number of Carmichael numbers not exceeding x it follows that the series Σ_n=1^∞1/C_n^1-ɛ is divergent for every ɛ>0.

Keywords:

Carmichael number , Lucas number , Lucas pseudoprime

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Rotkiewicz, A. (2007). On Lucas numbers, Lucas pseudoprimes and a numbertheoretical series involving Lucas pseudoprimes and Carmichael numbers. Annales Mathematicae Silesianae, 21, 49–60. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14063

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Vol. 21 (2007)
Published: 2007-09-28

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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