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Prime constellations
Here, we make a distinction between a prime constellation and a prime cluster.[1]
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Prime clusters
A prime cluster, also called a prime k-tuple (double, triple, quadruple, ...) is a strictly increasing sequence of primes such that the difference between the first and last is not necessarily minimal, i.e. some primes could be skipped over. More precisely, a prime k-tuple is a strictly increasing sequence of primes with , where is not necessarily the smallest number for which there exist integers and for every prime , not all the residues modulo are represented by .[1]
Nonadmissible prime clusters
A nonadmissible prime cluster, also called a nonadmissible prime k-tuple, is a prime cluster such that for some prime , all the residues modulo are represented by . Only a finite number of nonadmissible prime clusters may appear at the beginning of a prime cluster sequence.
For each , this definition excludes a finite number of clusters at the beginning of the prime number sequence. For example, (97, 101, 103, 107, 109) satisfies the conditions of the definition of a prime 5-tuplet, but (3, 5, 7, 11, 13) does not because all three residues modulo 3 are represented.[1]
Prime constellations
A prime constellation, also called a prime k-tuplet (doublet, triplet, quadruplet, ...) is a maximally dense prime cluster, i.e. a sequence of consecutive primes, i.e. such that the difference between the first and last is minimal, i.e. no primes could be skipped over. More precisely, a prime -tuplet is a sequence of consecutive primes, i.e. with , where is the smallest number for which there exist integers and, for every prime , not all the residues modulo are represented by .[1]
Prime pairs
Twin primes
Twin primes: prime pairs |
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Cousin primes
Cousin primes: prime pairs |
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Prime triplets
Prime triplets: are all prime |
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Prime triplets: are all prime |
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Prime triplets: or are all prime |
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Prime quadruplets
Prime quadruplets: are all prime. |
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Prime 5 tuplets
Prime 5 tuplets: are all prime |
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Prime 5 tuplets: are all prime |
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Notes
- ↑ 1.0 1.1 1.2 1.3 Tony Forbes, Prime Clusters and Cunningham Chains, Mathematics of Computation, Volume 68, Number 228, pp. 1739-1747.
- ↑ Weisstein, Eric W., Twin Prime Conjecture, from MathWorld—A Wolfram Web Resource.
- ↑ 3.0 3.1 Weisstein, Eric W., k-Tuple Conjecture, from MathWorld—A Wolfram Web Resource.