Strong pseudoprimes to the first 9 prime bases

TL;DR

This paper proves a conjecture about the smallest strong pseudoprimes to the first 11 prime bases, confirming that a specific 19-digit number is the smallest such pseudoprime for these bases.

Contribution

The paper introduces algorithms that establish the exact value of the smallest strong pseudoprime to the first 11 prime bases, confirming a conjecture by Zhang.

Findings

Confirmed the conjecture that $ ext{psi}_9 = ext{psi}_{10} = ext{psi}_{11} = Q_{11}$.

Identified the exact smallest strong pseudoprime to the first 11 prime bases.

Validated the use of algorithms to determine pseudoprime properties.

Abstract

Define $\psi_m$ to be the smallest strong pseudoprime to the first $m$ prime bases. The exact value of $\psi_m$ is known for $1\le m \le 8$. Z. Zhang have found a 19-decimal-digit number $Q_{11}=3825\,12305\,65464\,13051$ which is a strong pseudoprime to the first 11 prime bases and he conjectured that $$\psi_9=\psi_{10}=\psi_{11}=Q_{11}.$$ We prove the conjecture by algorithms.