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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 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.
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By formula | |
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By integer sequence | |
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By property | |
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Base-dependent | |
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Patterns | k-tuples |
- Twin (p, p + 2)
- Triplet (p, p + 2 or p + 4, p + 6)
- Quadruplet (p, p + 2, p + 6, p + 8)
- Cousin (p, p + 4)
- Sexy (p, p + 6)
- Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...)
- Balanced (consecutive p − n, p, p + n)
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By size | |
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Complex numbers | |
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Composite numbers | |
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Related topics | |
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First 60 primes | |
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Possessing a specific set of other numbers |
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Expressible via specific sums |
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