Strong pseudoprimes to the first eight prime bases

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 has 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 tabulate all the strong pseudoprimes $n\le Q_{11}$ to the first eight prime bases, and prove Zhang’s conjecture.