**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

andreasonably conjecturethat there could be at least 2^{24 }( = 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 *W*_{0}** **be the smallest Williams number, then for integers *N*** < W_{0}** only one iteration of the OPQBT (see [Zhang
2002])
will decide whether

2020-12-06

2022-1-28