The Distribution of Lucas and Elliptic Pseudoprimes

Abstract

Abstract. Let ¿?(x) denote the counting function for Lucas pseudoprimes, and 2?(x) denote the elliptic pseudoprime counting function. We prove that, for large x , 5?(x) < xL(x)~l/2 and W(x) < xL(x)~l/3 , where L(x) = exp(logxlogloglogx/log logx).

On Assessing the Performance of Randomized Algorithms

“…and thus we can infer that p f N → 0 The bound (3.13) was improved by Gordon and Pomerance [9]. In the same paper they consider Lucas pseudoprimes (for a definition see, e.g., [2]) and they prove The interpretation of this result is easy to grasp: when the odd integer n is randomly picked in 1 10 12 , n satisfies the Fermat congruence (3.10) only if n is a prime or a pseudoprime to base 2.…”

On Types of Elliptic Pseudoprimes

“…In this section we give some background on elliptic pseudoprimes in general. For other articles that study elliptic pseudoprimes and related notions see [4,6,7,11,12,16,20,23,24,25]. Elliptic pseudoprimes are analogous to Fermat pseudoprimes, which are composites N for which…”

On Fibonacci numbers which are elliptic Carmichael

“…Obviously, there might be other aspects of the CM condition for d = 1 which we have overlooked and which may be invoked to prove that the set of n for which F n is ECarmichael is of asymptotic density zero, but we leave such a task to the reader. Finally, we point out that several authors have treated the more coarse notion of an P ∈ E pseudoprime, which is a composite integer n such that (n − a n + 1)P = O p for all p | n and a fixed P ∈ E(Q) of infinite order (see [5], [6], [7]), and proved that they are of asymptotic density zero. It makes sense to ask the same question for the set of n such that F n is an P ∈ E pseudoprime, but we have no idea how to attack this question.…”