Evaluation of the Effectiveness of the Frobenius Primality Test

TL;DR

This paper evaluates the Frobenius primality test, demonstrating its high reliability and proving it does not fail for numbers below 2^64, with implications for primality testing accuracy.

Contribution

The paper introduces a version of the Frobenius test and proves its correctness for numbers less than 2^64, providing new insights into its reliability.

Findings

No counterexamples to the Frobenius test found below 2^64

Frobenius pseudoprimes must have a large prime divisor (>3000)

The test is very rarely wrong in practice

Abstract

The Frobenius primality test is based on the properties of the Frobenius automorphism of the quadratic extension of the residue field. Although it is probabilistic, we show that is "very rarely wrong". To date there are no counterexamples to this method and there are reasons to believe that they do not exist at all. In this paper, we suggest a version of the Frobenius test and prove that it does not fail for numbers less than $2^{64}$. We also show that a "Frobenius pseudoprime" will necessarily have a prime divisor greater than 3000.