Infinitely Many Carmichael Numbers for a Modified Miller-Rabin Prime Test

TL;DR

This paper introduces a modified Miller-Rabin primality test, demonstrating it has infinitely many Carmichael numbers and exploring their growth and patterns through empirical analysis.

Contribution

It defines a new variant of the Miller-Rabin test that is intermediate in strength and proves the existence of infinitely many Carmichael numbers for this test.

Findings

The test has infinitely many Carmichael numbers.

Empirical results reveal growth patterns of these numbers.

A strong pattern in the distribution of Carmichael numbers is identified.

Abstract

We define a variant of the Miller-Rabin primality test, which is in between Miller-Rabin and Fermat in terms of strength. We show that this test has infinitely many "Carmichael" numbers. We show that the test can also be thought of as a variant of the Solovay-Strassen test. We explore the growth of the test's "Carmichael" numbers, giving some empirical results and a discussion of one particularly strong pattern which appears in the results.