Williams prime

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A Williams number is a natural number of the form (b-1)•bn-1 for integers b ≥ 2 and n ≥ 1.

A Williams prime is a Williams number which is prime.


Varying both signs, there're four different types of numbers similiar to Williams numbers.

Lists of primes for bases b and n-values can be found here:

Type Category List
table [1]
smallest [2]
MM: (b-1)•bn-1 here here here
MP: (b-1)•bn+1 here here here
PM: (b+1)•bn-1 here here here
PP: (b+1)•bn+1 here here here [3]


  1. The list table of any type contains only bases which are included as a separate page.
  2. The list of smallest primes of any base is an ASCII file for 2 ≤ b ≤ 1024. Any unknown value is given with the search range in brackets or empty.
  3. Values for bases b == 1 mod 3 are always divisible by 3, so not listed here.

External links

Number classes
General numbers
Special numbers
Prime numbers