Conjectures on strong psp

Back Ground

Define ψ_m to be the smallest strong pseudoprime to all the first m prime bases. …Denote by ψ’_m (resp. ψ’’_m) the smallest K2-( resp. C3-) spsp to all the first m prime bases… In details

Conjectures on ψ_{m （}9 ≤m ≤20)

In [Zhang and Tang 2003, Zhang 2005, Zhang 2007], we reasonably made the following

Conjecture 1. [Zhang and Tang 2003] We have ψ₉ =ψ₁₀ =ψ₁₁ = N₉=N₁₀=N₁₁ = 3825 12305 65464 13051.

Conjecture 2. [Zhang 2005] We have ψ₁₂ =ψ’₁₂ < 10²⁴ <ψ’’₁₂ , where

ψ’₁₂ = N₁₂ = 399165290221 * 798330580441 = 3186 65857 83403 11511 67461 (24 digits)

was found in [Zhang 2001a].

Conjecture 3. [Zhang 2007] We have ψ_m =ψ’_m < ψ’’_m for all m ≥ 12, or more precisely, we have

ψ₁₂ = 3186 65857 83403 11511 67461 (24 digits)；

ψ₁₃ = 1287836182261 * 2575672364521 = 33170 44064 67988 73859 61981 (25 digits) ;

ψ₁₄ = 54786377365501 * 109572754731001 = 600 30942 89670 10580 03125 96501 (28 digits) ;

ψ₁₅ = 172157429516701 * 344314859033401= 5927 63610 75595 57326 34463 30101 (29 digits) ;

ψ₁₆ = ψ₁₇ = 531099297693901 * 1062198595387801 = 56413 29280 21909 22101 40875 01701 (30 digits);

ψ₁₈=ψ₁₉ = 27778299663977101 * 55556599327954201 = 1543 26786 44434 20616 87767 76407 51301 (34 digits);

ψ₂₀ > 10³⁶. Note that Conjecture 3 covers Conjecture 2, which is the case m = 12.

See also http://mathworld.wolfram.com/StrongPseudoprime.html; http://primes.utm.edu/glossary/page.php?sort=Pseudoprime

Updated:

(1)In 2014, Y. Jiang and Y. Deng [Math. Comp. 83:290 (2014), 2915–292] proved Conjecture 1.

(2)In 2017, J. Sorenson and J. Webster [Math. Comp. 863:304 (2017), 985–1003.] proved Conjecture 3 on ψ_m for m=12 and 13.

CHALLENGE： I offer a prize of $50 to the first person who disproves Conjecture 3 for either m: 14 ≤m ≤20; i.e., I offer a prize of $50 to the first person who communicates to me a composite N <ψ’_m which is a strong pseudoprime to all the first m prime bases for either m: 14 ≤m ≤19; or a composite N < 10³⁶ which is a strong pseudoprime to all the first 20 prime bases. More clearly speaking, I will offer a prize of $50 to Alice who is the first person to communicate to me a composite N <ψ’₁₄ which is a strong pseudoprime to all the first 14 prime bases; I will also offer a prize of $50 to Bob who is the first person to communicate to me a composite N <ψ’₁₅ which is a strong pseudoprime to all the first 15 prime bases; … , I will also offer a prize of $50 to Pete who is the first person to communicate to me a composite N < 10³⁶ which is a strong pseudoprime to all the first 20 prime bases. (so, in total, $50 * (20-13)= $350 is waiting for you)

Claimants must state the prime factorization of any numbers submitted.

2020-12-06 2022-01-28