Prime Constellations are more constant than the stars in the sky.
Even after the closest stars burn out and become dark, mathematical
structure will exist. For example 1+1=2 is a timeless
mathematical truth. This fact will always be so. Look at this link about Goldbach's Comet. Also interesting Tao-Green Theorem March 2021 by Matt C. Anderson email: matthewcharlesanderson2@gmail.com
A table of the first few prime numbers contains data that has a
certain mathematical property. Similarly, a table of twin primes
also contains some mathematical truth. To me, it is interesting to
develop these tables. The University of Tennessee at Martin
currently has two web pages for primes and twin primes.
specifically
and
Table of contents
Prime numbers are positive integers that are divisible only by
one and themselves. The first few prime numbers are 2, 3, 5, 7,
11, 13 and 17. There are many unsolved problems regarding prime
numbers. There may be a hidden pattern that describes the distribution
of the primes. Goldbach's conjecture
states that every even integer greater than 2 is the sum of two
prime numbers. It has not been shown to be true for certain,
although the numerical evidence supports it. The Hardy Littlewood
second conjecture states that the densest set of n prime numbers is the
first n prime numbers. This can be written in symbols π(x+n) - π(x) is less than or equal to π(n) π(n) is the prime counting function. The wikipedia page on prime k-tuples is a good introduction. Additionally, Tony Forbes keeps a record of primes in various constellation patterns, Jens Krus Anderson keeps a nice webpage about prime numbers at http://primerecords.dk/ and Wolfram's Mathworld has a nice article about prime constellations.
Each constellation of a given length has a fixed number of
patterns. The list of number of patterns for a given k-tuplet can
be found in the Online Encyclopedia of Integer Sequences (OEIS), index A083409.
The k-tuple conjecture is also called the first Hardy and
Littlewood conjecture. Being a conjecture, it is an unproven
guess. It states that every admissable pattern gives rise to an
infinite number of primes and the asymptotic density of theses primes
can be calculated. An admissible pattern is usually written with
the first number as zero. Then the pattern is admissible if it
does not contain a trivial divisibility that prevents an infinite set of
primes that fit into that pattern. For example, any admissable
pattern that starts with 0 cannot contain an odd number. Because
there is only one even prime and all the rest are odd. Similarly,
three consecutive odd numbers is not an admissable set because one of
divisbility by 3 considerations. To check a set of length k, one
must test all the entries in the potentialy admissable set against the
primes less than k to makes sure all the residue classes are not filled
up. There is a nice computer program that will test a set for
admissability, and it can be found here.
N.J.A. Slone = NJAS Warut Roonguthai = WR Matt C. Anderson = MCA Tim Johannes Ohrtmann = TJO The Online Encyclopedia of Integer Sequences has prime lists for several patterns:
A001359 Initial member of twin primes pattern (0,2) enumeration count 100,000 author NJAS
A022004 Initial member of prime triple (0,2,6) enumeration count 10,000 author WR
A022005 Initial member of prime triple (0,4,6) enumeration count 10,000 author WR
A007530 Initial member of prime quadruple (0,2,6,8) enumeration count 10,000 author WR
A022006 Initial member of prime 5-tuplet (0,2,6,8,12) enumeration count 10,000 author WR
A022007 Initial member of prime 5-tuplet (0,4,6,10,12) enumeration count 10,000 author WR
A022008 Initial member of prime 6-tuplet (0,4,6,10,12,16) enumeration count 1,000 author WR
A022009 Initial member of prime 7-tuplet (0,2,6,8,12,18,20) enumeration count 10,000 author MCA
A022010 Initial member of prime 7-tuplet (0,2,8,12,14,18,20) enumeration count 10,000 author MCA
A022011 Initial member of prime 8-tuplet (0,2,6,8,12,18,20,26) enumeration count 10,000 author WR
A022012 Initial member of prime 8-tuplet (0,2,6,12,14,20,24,26) enumeration count 10,000 author WR
A022013 Initial member of prime 8-tuplet (0,6,8,14,18,20,24,26) enumeration count 10,000 author WR
A022545 Initial member of prime 9-tuplet (0,2,6,8,12,18,20,26,30) enumeration count 10,000 author WR
A022546 Initial member of prime 9-tuplet (0,2,6,12,14,20,24,26,30) enumeration count 10,000 author MCA
A022547 Initial member of prime 9-tuplet (0,4,6,10,16,18,24,28,30) enumeration count 10,000 author WR
A022548 Initial member of prime 9-tuplet (0,4,10,12,18,22,24,28,30) enumeration count 10,000 author WR
A027569 Initial member of prime 10-tuplet (0,2,6,8,12,18,20,26,30,32) enumeration count 10,000 author WR
A027570 Initial member of prime 10-tuplet (0,2,6,12,14,20,24,26,30,32) enumeration count 10,000 author WR
A213646 Initial member of prime 11-tuplet (0,4,6,10,16,18,24,28,30,34,36) enumeration count 6923 author MCA
A213647 Initial member of prime 11-tuplet (0,2,6,8,12,18,20,26,30,32,36) enumeration count 6800 author MCA
A213645 Initial member of prime 12-tuplet (0,2,6,8,12,18,20,26,30,32,36,42) enumeration count 2807 author MCA A213601 Initial member of prime 12-tuplet (0,6,10,12,16,22,24,30,34,36,40,42) enumeration count 2952 author MCA A234947 Initial members of prime 13-tuplet (0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48) enumeration count 854 author MCA D.
Jacobson has written some software that will calculate even longer
lists of prime constellations which are a subset of
the k-tuples. A257137 Initial members of prime 13-tuplet (0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48) enumeration count 944 author TJO A257138 Initial members of prime 13-tuplet (0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36, 46, 48) enumeration count 802 author TJO A257139 Initial members of prime 13-tuplet (0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48) enumeration count 817 author TJO A257140 Initial members of prime 13-tuplet (0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48) enumeration count 1036 author TJO A257141 Initial members of prime 13-tuplet (0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48) enumeration count 803 author TJO Tim Johannes Ohrtmann posted some primes for the OEIS for 14 to 17-tuplets. A257167 Initial members of prime 14-tuplet (0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 and 50 ) enumeration count 185 author TJO A257168 Initial members of prime 14-tuplet (0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 and 50 ) enumeration count 209 author TJO A257304 Initial members of prime 15-tuplet (0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 and 56) enumeration count 16 author TJO A257304 only has 8 calculated primes as of 9/22/2015. A257369 Initial members of prime 16-tuplet (0,4,6,10,16,18,24,28,30,34,40,46,48,54,58, 60) enumeration count 30 author TJO A257370 Initial members of prime 16-tuplet (0,2,6,12,14,20,26,30,32,36,42,44,50,54,56,60) enumeration count 37 author TJO A257374 Initial members of prime 17-tuplet (0,4, 10,12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66) enumeration count 2 author TJO A257375 Initial members of prime 17-tuplet (0,4,6,10,16,18,24,28,30,34,40,46,48,54,58,60,66) enumeration count 6 author TJO A257376 Initial members of prime 17-tuplet (0,6,8,12,18,20,26,32,36,38,42,48,50,56,60,62,66) enumeration count 4 author TJO A257377 Initial members of prime 17-tuplet (0,2,6,12,14,20,24,26,30,36,42,44,50,54,56,62,66) enumeration count 8 author TJO Tony Forbes has a nice website with prime constellations and world record large k-tuplets. and note - A213601 has a list of 73 primes and represents over 8 months of Maple computer calculation as of 8/23/2012.
A008407 gives the minimum width of a constellation of given length. Theory
Two concepts are important for the theory of prime constellations. The first is the concept of primorial.
k primorial (written k#) is the product of the first k primes. Let pk be the kth prime
The numbers in the third column are the product of the primes in the second column.
The second concept is part of group theory. The
multiplicative group of integers modulo n must be understood. Here
is the wikipedia article about this multiplicative group.
The notation reads (Z/nZ)*. It contains a subset of the integers
from 1 to n-1. The elements of (Z/nZ)* are the integers from 1 to
n-1 that are relatively prime to n. If n is a prime number, then
(Z/nZ)* contains all the integers from 1 to n-1. If n
has many divisors, then (Z/nZ)* will contain fewer elements.
One way to find these prime k-tuplets is to consider the
multiplicative group of integers mod k primorial. This group contains
the set of integers less than k primorial that are relatively prime to k
primorial.
2# = 6
The multiplicative group mod 6 has two elements.
(Z/6Z)* = {1,5}
This tells us that all primes greater than 3 have the form 6n +/- 1.
So if one wants to search for the smaller of twin prime pairs, one should look at numbers of the form 6*n+5.
It would be a waste of time to test even numbers for primality, because all primes greater than 2 are odd.
Similarly, the group mod 30 has 8 elements.
(Z/30Z)* = {1,7,11,13,17,19,23,29}
By looking at the differences between adjacent elements in this set, we can see where the 3 tuplets can be found.
prime triples of the form (p,p+2,p+6) can be found only in the expressions 30*n+11 and 30*n+17.
Also, constellations of length 4, which are similarly called 4-tuplets, have the pattern (p, p+2,p+6,p+8).
Close inspection of the set {1,7,11,13,17,19,23,29} gives that the
4-tuplets must have the form 30*n+11. So the next step is to loop
with the variable n and test for all the elements in the pattern being
prime.
The ordered set (Z/30Z)* = {1,7,11,13,17,19,23,29} can be manipulated by taking the differences between adjacent elements.
d30 = [6,4,2,4,2,4,6]
This shows that the pattern (p,p+2,p+6,p+8), which has differences [2,4,2], can be found inside the ordered set d30.
The file below ktpatt.txt was copied from Anthony Forbe's
website, and shows the patterns for constellations up to length
50.
This page was written by Matt Anderson. My email address is
matthewcharlesanderson2@gmail.com Also, see - Our page on 2-tuples Although
the search for factors of large numbers is not as good as the
Folding__@__Home project by Stanford University, which could save a
life, it is still fun. (worth a look) try the Google words (search) folding at home and here is the link This project is more than 16 years old.
References Weisstein, Eric W. "k-Tuple Conjecture." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/k-TupleConjecture.html Caldwel, Chris K "Prime k-tuple conjecture" From The Prime Glossary http://primes.utm.edu/glossary/xpage/PrimeKtupleConjecture.html | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Originaltext
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