Anomalous primes and the elliptic Korselt criterion

Abstract

We explore the relationship between elliptic Korselt numbers of Type I, a class of pseudoprimes introduced by Silverman in [20], and anomalous primes. We generalize a result in [20] that gives sufficient conditions for an elliptic Korselt number of Type I to be a product of anomalous primes. Finally, we prove that almost all elliptic Korselt numbers of Type I of the form n = pq are a product of anomalous primes.In Section 2, we prove a generalization of [20, Proposition 17]. We show that if n = p 1 · · · p m is a squarefree Type I elliptic Korselt number with p 1 < · · · < p m and √ pm 4 m ≤ p 1 · · · p m−1 ≤ 4 m , then p m is anomalous and a n = 1. Hence all but an even number of the primes p i are anomalous, and if p i is not anomalous, then a p i = −1. Furthermore, we note an error in the proof of [20, Proposition 17], providing a counterexample and the corrected statement. In particular, we show that if n = pq is an elliptic Korselt number of Type I with p < q for E/Q and 13 ≤ p ≤ √ q/16, then p and q are anomalous for E.

On Types of Elliptic Pseudoprimes

“…Using these criteria, we show that strong elliptic Carmichael numbers are also Euler elliptic Carmichael numbers when applicable (Corollary 4.13). In Section 4 we investigate the elliptic Korselt numbers of Type I introduced in [31] and show the following result which proves Conjecture 4.9 from [3].…”

“…In [3,Proposition 4.3] the authors show that products of distinct anomalous primes for an elliptic curve E/Q are elliptic Korselt numbers of Type I for E. Here, we consider the question of how often an elliptic Korselt number of Type I is a product of distinct anomalous primes. We prove the following conjecture from [3], which deals with the case in which the number in question is semiprime. 6.1.…”

“…Rather, P should be understood as (84, 448). Moreover, [25] indicates that N +1 2 3 P is congruent to (2345, 0) modulo N , but it seems to be congruent to (30041, 29274) modulo N . Note that 29274 is divisible by 17 and 41, but not 47, so N is not a strong elliptic pseudoprime for (E, P ).…”

“…where u is some unit in the field. In general, p splits in Q( 2,3,7,11,19,43, 67, 163}. Up to isomorphism over Q, there are only thirteen elliptic curves with CM by an order in one of these fields.…”

“…, for m ≥ 3. Given a point P = (x, y) = [x : y : 1] on E/k and a nonnegative integer n, the projective coordinates of nP are given as nP = φ n ψ n : ω n : ψ 3 n , where…”