Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 658.10003
Autor: Erdös, Paul; Kiss, P.; Sárközy, A.
Title: A lower bound for the counting function of Lucas pseudoprimes. (In English)
Source: Math. Comput. 51, No.183, 315-323 (1988).
Review: Let R be a nondegenerate Lucas sequence, i.e. a sequence R = (rn)oon = 0 defined by the recurrence rn = arn-1- brn-2 for n > 1, r0 = 0, r1 = 1, for fixed integers a and b with ab\ne 0, (a,b) = 1 and \alpha /\beta is not a root of unity, where \alpha and \beta are the roots of x2-ax+b = 0.
An odd composite positive integer n with (n,b) = 1 that divides the element rn-(D/n) of R (here D: = a2-4b and (D/n) is the Jacobi-symbol) is called a Lucas pseudoprime with respect to the sequence R, Lpp/R for short.
In this paper it is shown that the number R(x) of Lpp/R not exceeding x is bounded below by \exp{(log x)c} for sufficiently large x and absolute constant c. This improves a result of P. Kiss [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 28(1986), 153-159 (1985; Zbl 613.10008)]. For the constant c no explicit value is given.
Reviewer: R.J.Stroeker
Classif.: * 11A15 Power residues, etc.
11A25 Arithmetic functions, etc.
11B37 Recurrences
Keywords: lower bound; counting function; Lucas sequence; recurrence; Lucas pseudoprime
Citations: Zbl 613.10008
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