Prime Constellations
By Matt Anderson4/9/2011
Prime numbers are integers that are divisible by only 1 and themselves.P={primes} = {2,3,5,7,11,…}There are an infinite number of prime numbers.
Letπ(x) be the prime counting function.π(x) counts the number of primes less than or equal to x.
The prime number theorem states thatπ(x) grows like x/Ln(x).Specifically,
A prime k-tupleis an ordered set of values representing a repeatable pattern of prime numbers.Examples Instances(0,2) twin primes {3,5},{5,7},{11,13}(0,4) cousin primes {3,7},{7,11},{13,17}(0,6) sexy primes {5,11},{7,13},{11,17}(0,2,6) 3-tuple {3,5,11},{5,7,13}(0,4,6) 3-tuple {7,11,13},{13,17,19}(0,2,6,8) 4-tuple {5,7,11,13},{11,13,17,19}
A k-tupleis said to be admissible if it does not include the complete modulo set of residue classes (iethe values 0 through p-1)of any primep≤k.The k-tupleslisted thus far are all admissible, but (0,2,4) is not admissible. Since0 mod 3 = 02 mod 3 = 24 mod 3 = 1The complete set of residue classes mod 3 is {0,1,2}.The only primes that satisfy this 3-tuple are {3,5,7}.If the smallest prime is greater than 3 then it will not be possible for all three members to be prime.
An admissible prime k-tuplethat is maximally dense is called a constellation with k primes.Forn≥k, this will always produce consecutive primes. Where, n is the smallest prime in the constellation.Example: The constellation with 2 primes is (0,2).Example 2: There are 2 constellations with 3 primes.They are (0,2,6) and (0,4,6).
It is conjectured that there are an infinite number of twin primes. Also, it is conjectured that there are an infinite number of primes for every admissible k-tuple.Numerical evidence supports this conjecture.
Letπm1, m2,…,mk(x) be the number of(k+1)tuplesless than xFor example, consider the4-tuple (0,2,6,8)The smallest prime has the form 30n+11.
For exampleπ2(x) counts the number of twin primes less than or equal to x. Similarlyπ4(x) counts the number of cousin primes less than or equal to x. Twin primes have the from 6x+5 and 6x+7. We know that from divisibility by 2 and 3.Primes p > 3 must have the form:p=1 mod 6orp=5 mod 6otherwise, they would be divisible by 2 or 3.
To find k-tuples, one must determine the values of a and b inp=ax+bfor the smallest prime p in a constellation. One way to do this is by examining Ur# where r# (read rprimorial) is the product of the first r primes and Ur# is the set of units mod r#.
For example:U2 = {1}. All prime numbers greater than 2 are odd.U6 = {1,5}. All primes > 3 have the form 6k±1U30 = {1,7,11,13,17,19,23,29}
The first HardyLittlewoodConjecture states that every admissible (k+1)-tuplehas infinitely many prime examples and the asymptotic distribution is given by:πm1,m2,…,mk(x) ~ C(m1,m2,…,mk)and w(q;m1,m2,…mk) is the number of distinct residues of m1, m2,…,mk(mod q)
The second HardyLittlewoodconjecture states that:π(x+y) ≤ π(x) + π(y)for all x, y with 2 ≤ x ≤ y.It is believed that there is a counterexample for x=447 and 10147< y < 101199
Examples have been found for constellations of 2 to 23 primes. There are no known examples of constellations with 24 primes.
References:http://www.sam.math.ethz.ch/~waldvoge/Projects/clprimes05.pdfhttp://www.opertech.com/primes/k-tuples.htmlhttp://anthony.d.forbes.googlepages.com/ktuplets.htmhttp://mathworld.wolfram.com/Hardy-LittlewoodConjectures.htmlElementary Number Theory 2ndEdition by Underwood DudleyPrime Numbers: A Computational Perspective 2ndEditionby Richard Crandall and CarlPomerance
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