Lucas pseudoprimes

Abstract

We define several types of pseudoprimes with respect to Lucas sequences and prove the analogs of various theorems about ordinary pseudoprimes. For exam

The Rabin-Monier theorem for Lucas pseudoprimes

“…As above, we can derive a "test" from this property: the strong Lucas pseudoprime test [4]. In this test, we check whether property (2) holds, for several pairs (P, Q).…”

“…Among numbers n such that SL(D, n) does to exceed 4 15 n, consider those such that n = p 1 p 2 p 3 ≡ 1 modulo 4, ε(p i ) = −1, and p i + 1|n + 1 for i = 1, 2, 3 (these numbers were already encountered in [10]). We have, in this case, SL(D, n) = (q 1 − 1)(q 2 − 1)(q 3 − 1) + q 1 q 2 q 3 which can be greater than n/4, and very close to 4/15n.…”

“…There exist several ways to improve the Lucas test in order to make it more secure. One good idea yet found in [4] and [8] is to combine a Rabin-Miller test and a "true" (i.e. with (D/n) = −1) Lucas test.…”

Statistical Evidence for Small Generating Sets

“…For example, a result of Baillie and Wagstaff [2,Theorem 9] shows that when n is odd, L(n), the least prime /? with (p\n) ^ 1, has a mean value in accordance with probability theory, provided one accounts for the primes dividing « .…”

The Distribution of Lucas and Elliptic Pseudoprimes

“…The upper bound is due to Baillie and Wagstaff [1], and the lower bound is due to Erdös, Kiss, and Sarközy [5]. Of course, the counting function J¿?…”