The non-existence of universal Carmichael numbers

TL;DR

This paper proves that universal elliptic Carmichael numbers do not exist and provides probabilistic bounds on the occurrence of elliptic Carmichael numbers for random integers and elliptic curves.

Contribution

It establishes the non-existence of universal elliptic Carmichael numbers and derives probabilistic bounds for elliptic Carmichael numbers for random integers and elliptic curves.

Findings

Universal elliptic Carmichael numbers do not exist.

Probability that a non-prime power integer is elliptic Carmichael for a random curve is bounded by O(log^{-1} n).

Probability that a random integer and curve pair yields an elliptic Carmichael number is bounded by O(n^{-1/8+ε}).

Abstract

We show that universal elliptic Carmichael numbers do not exist, answering a question of Silverman. Moreover, we show that the probability that an integer $n$, which is not a prime power, is an elliptic Carmichael number for a random curve $E$ with good reduction modulo $n$, is bounded above by $\mathcal{O}(\log^{-1} n)$. If we choose both $n$ and $E$ at random, the probability that $n$ is $E$-carmichael is bounded above by $\mathcal{O}(n^{-1/8+\epsilon})$.