Williams prime
Definition
A Williams number is a natural number of the form (b1)•b^{n}1 for integers b ≥ 2 and n ≥ 1.
A Williams prime is a Williams number which is prime.
Generalization
Varying both signs, there're four different types of numbers similiar to Williams numbers.
Lists of primes for bases b and nvalues can be found here:
Type  Category  List table ^{[1]} 
List smallest ^{[2]} 

MM: (b1)•b^{n}1  here  here  here 
MP: (b1)•b^{n}+1  here  here  here 
PM: (b+1)•b^{n}1  here  here  here 
PP: (b+1)•b^{n}+1  here  here  here ^{[3]} 
Notes
 ↑ The list table of any type contains only bases which are included as a separate page.
 ↑ 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.
 ↑ Values for bases b == 1 mod 3 are always divisible by 3, so not listed here.
External links
 H. C. Williams: "The primality of certain integers of the form 2Ar^n1", Acta Arith. 39 (1981), 717
 A. Stein, H. C. Williams: "Explicit primality criteria for (p−1)p^{n}−1", Math. Comp. 69 (2000), 17211734
 Steven Harvey: Search for original Williams primes
 Mauro Fiorentini: Type MM, Type MP, Type PM, Type PP for 0 ≥ n ≥ 1000 (mostly) and 1 ≥ b ≥ 1000 (2016)
 Williams number
Number classes
General numbers 
Special numbers 
Prime numbers 
