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Williams prime

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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[1] Table [2] Smallest [3] Remaining[4]
MM: (b-1)bn-1 here here
169 bases
here[5]
40 unknown
here
MP: (b-1)bn+1 here here
154 bases
here
21 unknown
here
PM: (b+1)bn-1 here here
87 bases
here
1 unknown
here
PP: (b+1)bn+1 here here
82 bases
here [6]
4 unknown
here
1. Containing all related pages for the type.
2. The table contains only bases which are included as a separate page.
3. The list of smallest primes of any base is an ASCII file for 2 ≤ b ≤ 1024. For unknown values only the base is given.
4. All data not yet available as separate page.
5. The list contains values for 2 ≤ b ≤ 2049.
6. Values for bases b ≡ 1 mod 3 are always divisible by 3, so not listed here.

Available Online Sequences

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