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Class of numbers in number theory
In number theory , a Williams number base b is a natural number of the form
(
b
−
1
)
⋅
b
n
−
1
{\displaystyle (b-1)\cdot b^{n}-1}
for integers b ≥ 2 and n ≥ 1.[ 1] The Williams numbers base 2 are exactly the Mersenne numbers .
A Williams prime is a Williams number that is prime . They were considered by Hugh C. Williams .[ 2]
It is conjectured that for every b ≥ 2, there are infinitely many Williams primes for base b .
See also
References
External links
By formula By integer sequence By property Base -dependentPatterns
Twin (p , p + 2 )
Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, … )
Triplet (p , p + 2 or p + 4, p + 6 )
Quadruplet (p , p + 2, p + 6, p + 8 )
k -tuple
Cousin (p , p + 4 )
Sexy (p , p + 6 )
Chen
Sophie Germain/Safe (p , 2p + 1 )
Cunningham (p , 2p ± 1, 4p ± 3, 8p ± 7, ... )
Arithmetic progression (p + a·n , n = 0, 1, 2, 3, ... )
Balanced (consecutive p − n , p , p + n )
By size Complex numbers Composite numbers Related topics First 60 primes
Possessing a specific set of other numbers
Expressible via specific sums