On Lucas numbers, Lucas pseudoprimes and a numbertheoretical series involving Lucas pseudoprimes and Carmichael numbers
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|αx + βx 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 Σn=1∞1/logPn(a)=1, where Pn(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|Pn(a))=ɛ, is divergent.
(3) Let Cn denote the n-th Carmichael number. From the conjecture of P. Erdős that C(x) > x1−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 Σn=1∞1/Cn1-ɛ is divergent for every ɛ>0.
Keywords
Carmichael number; Lucas number; Lucas pseudoprime
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Instytut Matematyki, Polska Akademia Nauk Poland
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