There are infinitely many elliptic Carmichael numbers

TL;DR

This paper proves the existence of infinitely many elliptic Carmichael numbers, extending the understanding of these special composite numbers in elliptic curve number theory.

Contribution

It establishes the infinitude of elliptic Carmichael numbers, answering a longstanding open question in the field.

Findings

Proved there are infinitely many elliptic Carmichael numbers.

Resolved the question of the infinitude of certain square-free, composite integers.

Connected elliptic Carmichael numbers to divisibility conditions on primes.

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 square-free, composite integers $n$ such that for every prime $p$ that divides $n$, $p+1|n+1$.