Strong Pseudoprimes to Twelve Prime Bases

TL;DR

This paper extends the known smallest strong pseudoprimes to twelve and thirteen prime bases and introduces an algorithm for identifying all such pseudoprimes up to a bound, with a heuristic analysis of its efficiency.

Contribution

It determines new values of the smallest strong pseudoprimes for the 12th and 13th prime bases and proposes an efficient algorithm for finding all strong pseudoprimes up to a given bound.

Findings

Determined $ ext{psi}_{12}$ and $ ext{psi}_{13}$ values.

Developed an algorithm with heuristic analysis for finding strong pseudoprimes.

Estimated the algorithm's running time as at most $B^{2/3+o(1)}$.

Abstract

Let $\psi_m$ be the smallest strong pseudoprime to the first $m$ prime bases. This value is known for $1 \leq m \leq 11$. We extend this by finding $\psi_{12}$ and $\psi_{13}$. We also present an algorithm to find all integers $n\le B$ that are strong pseudoprimes to the first $m$ prime bases; with a reasonable heuristic assumption we can show that it takes at most $B^{2/3+o(1)}$ time.