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:

• Type (b-1)•bⁿ-1
• Type (b-1)•bⁿ+1
• Type (b+1)•bⁿ-1
• Type (b+1)•bⁿ+1

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 original Williams primes
• Williams number

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