Primitive Lucas d-pseudoprimes and Carmichael–Lucas numbers

The Distribution of Lucas and Elliptic Pseudoprimes

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).

Carmichael numbers and pseudoprimes

We now establish a pleasantly simple description of Carmichael numbers, due to Korselt. First, we need the following notion. Let a and p be coprime (usually, p will be prime, but this is not essential). The order of a modulo p, denoted by ordp(a), is the smallest positive integer m such that a ≡ 1 mod p. Recall [NT4.5]: If ordp(a) = m and r is any integer such that a ≡ 1 mod p, then r is a mult...

Existence of Primitive Divisors of Lucas and Lehmer Numbers

We prove that for n > 30, every n-th Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor. Whether the mathematicians like it or not, the computer is here to stay. Folklore Whether the computer likes it or not, mathematics is here to stay. Beno Eckmann 32], p. xxiii Contents 1 Introduction 1 2 Cyclotomic criterion and n...

Pseudoprimes and Carmichael Numbers

The Rabin-Monier theorem for Lucas pseudoprimes

We give bounds on the number of pairs (P,Q) with 0 ≤ P,Q < n such that a composite number n is a strong Lucas pseudoprime with respect to the parameters (P,Q).