Infinitude of Elliptic Carmichael Numbers

Abstract

In 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat's little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author.2010 Mathematics subject classification: primary 11A51; secondary 11G05, 11Y11, 11N13, 11N36, 11A07.

There are infinitely many elliptic Carmichael numbers

“…Our task is made easier by a reformulation of the question (as noted in ), as the search for elliptic Carmichael numbers can be reduced to the following Korselt‐like criterion : Elliptic Carmichael Condition Let be a squarefree, composite positive integer with an odd number of prime factors. Moreover, let Then is an elliptic Carmichael number if for each prime , we have and .…”

On Fibonacci numbers which are elliptic Carmichael

“…In [5], the authors gave a sufficient condition for a positive integer n to be an elliptic Carmichael number for all E with CM by Q( √ −d) resembling the Korselt criterion for Carmichael numbers. In [8], we showed that the counting function of the set of n such that F n fulfills that criterion is O(x(log log x) 1/2 /(log x) 1/2 ).…”

“…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.…”

Carmichael Numbers in Arithmetic progressions

“…This seems very likely but difficult to prove. Thinking of G as ⊕ q∈Q (Z/φ(q)Z) ⊕ (Z/φ(m)Z), an example of a troublesome large subgroup would be Ekstrom et al [7] have developed the arguments in [2,5,6] further to give a conditional proof of the infinitude of elliptic Carmichael numbers. Unfortunately our methods do not seem to be directly helpful in that problem.…”