There are infinitely many elliptic Carmichael numbers

Abstract

In 1987, Dan Gordon defined an elliptic curve analogue to Carmichael numbers known as elliptic Carmichael numbers. In this paper, we prove that there are infinitely many elliptic Carmichael numbers. In doing so, we resolve in the affirmative the question of whether there exist infinitely many Lucas-Carmichael numbers (that is, squarefree, composite integers n such that for every prime p that divides n, p + 1|n + 1) .

A Conditional Density for Carmichael Numbers

“…This conjecture was first used by Banks, Ekstrom, Pomerance and Thakur in two papers [2,4] that would prove (conditionally) that (a) there are infinitely many Carmichael numbers in arithmetic progressions (if the modulus and residue class are coprime) and (b) there are infinitely many composite square-free numbers m such that p + 1 | m + 1 for any prime p which divides m. Although these results were later made unconditional in [16] and [14], the conjecture has also been used in other Carmichael results, including [15].…”

There are infinitely many (-1,1)-Carmichael numbers

“…The ideas came to us after seeing the papers [1] and [8]. The method we used in this paper is a simple modification of method in [1].…”

“…However, there are no any result about infinitude of (a, 1)-Carmichael numbers but a = 1. In 2018, Wright [8] proved that there are infinitely many (−1, −1)-Carmichael numbers, or Lucas-Carmichael numbers.…”

Factors of Carmichael Numbers and an Even Weaker -Tuples Conjecture

“…Note that while the proof above is roughly the same as the one in [1] or [14], we use it here for different purposes. In the cited works, this result is used to generate primes p that can be combined into Carmichael numbers.…”