Williams prime

Definition

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.

Generalization

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	Table	Smallest	Remaining
MM: (b-1)•bn-1	here	here 214 bases	here 40 unknown	here
MP: (b-1)•bn+1	here	here 189 bases	here 18 unknown	here
PM: (b+1)•bn-1	here	here 128 bases	here 2 unknown	here
PP: (b+1)•bn+1	here	here 110 bases	here 4 unknown	here

Available Online Sequences

Here are listed the available sequences in the On-Line Encyclopedia of Integer Sequences.

b	MM (b-1)•bn-1	MP (b-1)•bn+1	PM (b+1)•bn-1	PP (b+1)•bn+1
2	A000043 -> page		A002235 -> page	A002253 -> page
3	A003307 -> page	A003306 -> page	A005540 -> page	A005537 -> page
4	A272057 -> page	A326655 -> page
5	A046865 -> page	A204322 -> page	A257790 -> page	A143279 -> page
6	A079906 -> page	A247260 -> page
7	A046866 -> page	A245241 -> page
8	A268061 -> page	A269544 -> page
9	A268356 -> page	A056799 -> page
10	A056725 -> page	A056797 -> page	A111391 -> page
11	A046867 -> page	A057462 -> page
12	A079907 -> page	A251259 -> page
13	A297348 -> page
14	A273523 -> page

External links

• H. C. Williams: "The primality of certain integers of the form 2Ar^n-1", Acta Arith. 39 (1981), 7-17
• A. Stein, H. C. Williams: "Explicit primality criteria for (p−1)pⁿ−1", Math. Comp. 69 (2000), 1721-1734
• Steven Harvey: Search for Williams primes: only Type MM (b-1)•bⁿ-1 for 3 ≤ b ≤ 1024, 1 ≤ n ≤ 512 and 1025 ≤ b ≤ 2049, 1 ≤ n ≤ 100 and some higher (2006-2019)
• Mauro Fiorentini: Type MM, Type MP, Type PM, Type PP for 0 ≤ n ≤ 1000 (mostly) and 1 ≤ b ≤ 1000 (2016)
• Eric Chen: Thread at MersenneForum including dual forms for 0 ≤ n ≤ 5000 and 1 ≤ b ≤ 64 and some higher (2016-2019)
• Williams number

Williams primes

Main	Williams prime

Type MM: (b-1)•bn-1

Category Table (214) least n-values remaining data wanted values

Type MP: (b-1)•bn+1

Category Table (189) least n-values remaining data wanted values

Type PM: (b+1)•bn-1

Category Table (128) least n-values remaining data wanted values

Type PP: (b+1)•bn+1

Category Table (110) least n-values remaining data wanted values

Number classes

General numbers

Integer Natural Even Odd Rational Irrational Real Prime Probable prime

Special numbers

Mersenne 2n-1 Fermat 22n+1 Proth k•2n+1 Riesel k•2n-1 Woodall n•2n-1 Cullen n•2n+1 Double Mersenne 22p-1-1 Perfect 2n-1(2n-1) Leyland xy ± yx Gen.Repunit (b^n-1)/(b-1)

Prime numbers

Mersenne Proth Riesel Williams (b±1)bn±1 Carol/Kynea (bn±1)2-2 Multifactorial n!j±1