In the literature [1], Carmichael Numbers that satisfy additional constraints $(p+1) \mydivides (N+1)$ for every prime divisor $p \mydivides N$ are referred to as ``Williams' Numbers''\footnote{more precisely, ``1-Williams Numbers''~; however~; the distinctions between different types of Willliams' numbers are not relevant in this document and therefore, we refer to 1-Williams Numbers~ simply as Williams' numbers.}. % In the renowned Pomerance-recipe~\cite{pomerance1984there} to search for Baillie-PSW pseudoprimes; there are heuristic arguments suggesting that the number of Williams' Numbers could be large (or even unlimited). Moreover, it is shown~\cite{pomerance1984there} that if a Williams' number is encountered during a search in accordance with all of the conditions in that recipe~\cite{pomerance1984there}~; then it must also be a Baillie-PSW pseudoprime. We derive new analytic bounds on the prime-divisors of a Williams' Number.\\ Application of the bounds to Grantham's set of 2030 primes~(see ~\cite{grantham-620-list}) drastically reduces the search space from the impossible size $\approx 2^{(2030)}$ to less than a quarter billion cases (160,681,183 cases to be exact, please see the appendix for details). We tested every single case in the reduced search space with maple code. The result showed that there is \underline{NO Williams' number (and therefore NO Baillie-PSW pseudo-prime which is also a Williams' number)} in the entire space of subsets of the Grantham-set. The results thus demonstrate that Williams' numbers either do not exist or are extremely rare. We believe the former; i.e., that No such composite (i.e., a Williams' Number of this type) exists.