On the infinitude of elliptic Carmichael numbers

Abstract

For any elliptic curve E with complex multiplication by an order in K=Q(√-d), a point Q of infinite order on E, and any prime p with gcd{Δ(E),p} = 1, (-dǀp) = -1 , we have that [p+1]·Q = O (mod p) where Δ(E) is the discriminant of E, O is the point at infinity and calculations are done using the addition law for E. Any composite number p which satisfies these conditions for any rational point on any CM elliptic curve over Q with discriminant prime to p is called an elliptic Carmichael number. For our main result, we modify the techniques of Alford, Granville and Pomerance to show there exist infinitely many elliptic Carmichael numbers under the assumption that the smallest prime congruent to -1 modulo q is at most q exp[(log q)¹⁻ᵋ]. (Note that this assumption is much weaker than current conjectures about the smallest prime in an arithmetic progression.) We modify the construction of Chernick to provide many examples of elliptic Carmichael numbers. We also show that elliptic Carmichael numbers are squarefree, and modify the techniques of Pomerance, Selfridge, and Wagstaff to prove that the number of elliptic Carmichael numbers up to x with exactly k factors is at most x⁽²ᵏ⁻¹⁾/⁽²ᵏ⁾⁺ᵋ for large enough x. Finally, we prove there are no strong elliptic Carmichael numbers, an analogue of Lehmer's result about strong Carmichael numbers.