Constellation Primes

(Draft, under construction)

Objectives

This page lists small members of tuple primes of various patterns.

This page is not just a compilation of data obtained from the public, nor just verification of the data from different sources. Although some of the results in this page are already known and published in the literature or on the internet, the data presented in this page are entirely generated by various sieve programs written in C++ by me.

The actual time required to obtain various data is also generated during the execution of the sieve program. However, I have run these programs in different computers, and thus these values should not be compared with each other.

Note : As the tables are generally very large, esp. those for Twin primes and 3-tuple primes, I have not posted the raw data onto this web site.

Terminology

Before I am able to present the results of my sieve, let me define the terminlogies that I will be using. Some of these terms may be different from those defined by others elsewhere in the literature or on the internet. Among these differences, there is one most important difference in my usage here; that is, the formation of a k-tuple prime constellation does not require these prime numbers to be consecutive primes. For example, I consider the tuple (13,17,19,29) a 4-tuple prime constellation even though 23 is a prime number within the range 13 and 29.

Constellation primes

The term "constellation prime" is borrowed from Weisstein's World of Mathematics, but it is used in a more broader sense in this web page. In Weisstein's definition, the term "constellation prime" is equivalent to k-tuple prime, whereas k-tuple prime is regarded as only a special form of constellation primes.. Constellation Primes is here defined as a group of prime numbers which are related to each other by some form of functional relations.

A group of primes (p₁, p₂, ..., p_n) is said to form a prime constellation if they satisfy the following set of relations :

p_i = f_i(i)

Classifications

Twin Primes

Two prime numbers are said to be a Twin prime pair if their difference is 2.

Sophie Germain Primes

Two prime numbers are said to be a pair of Sophie Germain primes if p₂ = 2 p₁ + 1.

Stern Primes

Two prime numbers are said to be a pair of Stern primes if p₂ = 4 p₁ + 1.

Some properties of Stern primes in cryptogragphy can be found in p.169 in Stream Cipher and Number Theory by Cusick, Ding and Renvall.

k-Tuple Primes

A group of prime numbers (p₁, p₂, ...,p_k) is said to form a prime k-tuple if p_i = p_i-1 + a_i, for i = 2, 3,...,k.

Twin primes is a special form of 2-tuple primes.

Cunningham Chain

A group of prime numbers (p₁, p₂, ...,p_k) is said to form a Cunningham chain if p_i = 2 p_i-1 - 1, for i = 2, 3,...,k.

Dickson Primes

A group of prime numbers (p₁, p₂, ...,p_k) is said to form a k-tuple Dickson primes if there exists an integer n such that p_i = a_i n+ b_i, for i = 1, 2,...,k. Dickson primes are studied in the well-known Dickson's conjecture, ref. p.373 of The New Book of Prime Number Records written by Paulo Ribenboim. Sophie Germain primes and Cunningham chain are special forms of Dickson primes. k-Tuple primes is also a special form of Dickson primes with all a_i= 1.

Hua Primes

Hua primes is a group of prime numbers (p₁, p₂, ...,p_k) such that p_i = p₁ + i² - i, for i = 1, 2, ...,k.

Hua primes are generated according to Hua Loo Keng, p.74 of Introduction to Number Theory.

Type of Prime Constellation	Tuple Length	Generating relations
Twin Primes	2	p₂ = p₁ + 2
Sophie Germain Primes	2	p₂ = 2 p₁ + 1
Stern Primes	2	p₂ = 4 p₁ + 1
k-Tuple Primes	k	p_i = p_i-1 + a_i-1, for i = 2, 3,...,k
Cunningham Chain	k	p_i = 2 p_i-1 - 1, for i = 2, 3,...,k
Dickson Primes	k	p_i = a_i n + b_i, for i = 1,2,3,...,k
Hua Primes	k	p_i = p₁ + i² - i, for i = 1, 2, ...,k

Tuple length

Tuple length is defined as the number of primes in the prime constellation.

Tuple distance

Tuple distance is defined as the difference between the smallest and the largest of primes in in the prime constellation.

Selected Tables of Tuple Primes

Since the web hosting service does not allow enough space to post all the data in this web site, most of the tables do not contain the raw data. However, as some of these k-tuple primes will be interested to some readers, they are presented in Selected Tables of Tuple Primes.

Observations and Conjectures

With the results obtained from the sieve algorithms, some observations are noted. Particularly, two lemmas related to the k-tuple conjecture are derived. And it is hope that it can be used eventually to prove the k-tuple conjecture.

References

Eric W. Weisstein "k-Tuple Conjecture" http://mathworld.wolfram.com/k-TupleConjecture.html

Eric W. Weisstein "Prime Constellation" http://mathworld.wolfram.com/PrimeConstellation.html

Forbes, T. "Prime k-Tuplets" http://www.ltkz.demon.co.uk/ktmin.txt

First posted onto the internet : 24 September, 2003.