A Conjecture on Williams Numbers

In 1977, Williams asked whether there are any Carmichael numbers n with an odd number of prime divisors and the additional property that for each prime p | n,

p + 1 | n + 1.

We [Zhang 2015, §7] tabulated a table of 1029 primes: ( Download the text file of the 1029 primes )

787

907

1087

…

224489273360340970879

4194019937445612397579

4905729483343410746599

and reasonably conjecture that there could be at least 224  ( = 16,777,216)  Williams numbers made up from them. 
 

CHALLENGE： I offer a prize of $500 to the first person who communicates to me a Williams number made up from the 1029 primes.

Claimants must state the prime factorization of any numbers submitted.

(Let W₀ be the smallest Williams number, then for integers N < W₀ only one iteration of the OPQBT (see [Zhang 2002]) will decide whether N is prime or not. However， our prize does not ask you to submit W_{0 ,} only one Williams number is enough to win the prize of $500. Finding W₀ is the next project. )

2020-12-06

2022-1-28