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 )







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 W0 , only one Williams number is enough to win the prize of $500.  Finding W0  is the next project.  )