Strong pseudoprimes to the first eight prime bases

Abstract

Define ψ m to be the smallest strong pseudoprime to the first m prime bases. The exact value of ψ m is known for 1 ≤ m ≤ 8. Z. Zhang have found a 19decimal-digit number Q 11 = 3825 12305 65464 13051 which is a strong pseudoprime to the first 11 prime bases and he conjectured that ψ 9 = ψ 10 = ψ 11 = Q 11 . We prove the conjecture by algorithms.

Strong pseudoprimes to twelve prime bases

“…By looking at a narrow class of numbers, Zhang [18] gave upper bounds on ψ m for 9 ≤ m ≤ 19. Recently, Jian and Deng [8] verified some of Zhang's conjectures and computed ψ m for m = 9, 10, and 11. We continue in this effort with the following.…”

“…As a practical matter, these computations only need to be done once. As an example, [8] found 84 98355 74122 37221 = 206135341 · 412270681 to be a strong pseudoprime to the first eight prime bases. When considering p 1 = 206135341, they sieved (which we explain in the next section) for p 2 = 412270681.…”

“…The authors of [8] consider searching for p t modulo lcm{λ, 9240}. They considered 30 residue classes modulo 9240 = 8 · 3 · 5 · 7 · 11 that arose from the propositions above.…”

Tabulating Pseudoprimes and Tabulating Liars

“…In the process, upper bounds or exact values were given for ψ k with k ≤ 19. The authors of [18] tabulated pseudoprimes to the first eight prime bases and provably determined ψ 11 . An algorithm developed in [6] tabulates pseudoprimes to two or more bases up to x and has a heuristic complexity of O(x 9/11+o(1) ).…”