# A Probable Prime Test With High Confidence

## TL;DR

This paper introduces a new probable prime test using quadratic polynomials and Frobenius automorphism, achieving higher confidence by reducing the likelihood of composite numbers passing the test compared to previous methods.

## Contribution

A novel probable prime test based on quadratic polynomials and Frobenius automorphism that significantly lowers false positives for composite numbers.

## Key findings

- Composite passes less than 1/7710 of the polynomial tests
- Test runs approximately three times slower than the Strong Probable Prime Test
- Provides higher confidence in primality testing

## Abstract

Monier and Rabin proved that an odd composite can pass the Strong Probable Prime Test for at most $\frac 14$ of the possible bases. In this paper, a probable prime test is developed using quadratic polynomials and the Frobenius automorphism. The test, along with a fixed number of trial divisions, ensures that a composite $n$ will pass for less than $\frac 1{7710}$ of the polynomials $x^2-bx-c$ with $\left(b^2+4c\over n\right)=-1$ and $\left(-c\over n\right)=1$. The running time of the test is asymptotically $3$ times that of the Strong Probable Prime Test.