Ours go to 211: Euler pseudoprimes to 47 prime bases (from Carmichael numbers)

TL;DR

This paper identifies a subset of Carmichael numbers that serve as strong Euler pseudoprimes for many bases, introduces a fast algorithm to find such pseudoprimes, and discovers numbers surviving up to the 47th prime base.

Contribution

It classifies Carmichael numbers to efficiently generate Euler pseudoprimes, extending the known range of bases for which these pseudoprimes are valid.

Findings

Found Euler pseudoprimes surviving up to the 47th prime base.

Developed a fast algorithm to generate new Euler pseudoprimes.

Identified a subset of Carmichael numbers with strong pseudoprime properties.

Abstract

In this paper we show that a certain subset of the Carmichael numbers contains good Euler pseudoprimes, composite numbers that for many bases survive the Solovay-Strassen primality test. We present a classification of Carmichael numbers, and use the knowledge gained from this to create a fast algorithm to compute new Euler pseudoprimes, by multiplying already found Euler pseudoprimes. We use this algorithm to find many Euler pseudoprimes that are pseudoprimes for several consecutive prime bases starting at 2, hence for all integer bases up to that number. The best Euler pseudoprime we find survives up to 211, i.e., survives the first 47 prime bases.