Special Classes of Primes near Primorials

Special Classes

We will now look at relationships between some of the special classes of primes near a primorial and the corresponding primes.

Twin Primes

Primes within the proven primorial influence that are Twin Primes have corresponding primes that are also Twin Primes.

Theorem:

If there are twin primes above and near a primorial:

ap_n# + c, ap_n# + c + 2 where c > p_n and c+2 < p_n+1²

then

c and c+2 are twin primes.

If there are twin primes below and near a primorial:

ap_n# - c, ap_n# - c + 2 where c+2 > p_n and c < p_n+1²

then

c and c+2 are twin primes.

Other forms

This concept can be proved for other forms such as tuplets, cousin primes, and other numbers of primes of like classes.

Cunningham Chains

Primes near the primorial that are within the proven primorial influence that are Cunningham Chains have corresponding Cunningham Chains. If the primes are above the primorial, the Cunningham Chains are of the same kind. If the primes are below the primorial, the Cunningham Chains are of the opposite kind.

Definition

A Cunningham chain of the first kind of length n is a sequence of prime numbers (x₁,...,x_n) such that for all 1 ≤ i < n, x_i+1 = 2 x_i + 1.

Theorem

If there is a Cunningham Chain of the first kind where all values are under primorial influence and above the primorial, then there is also a corresponding Cunningham Chain of the first kind.

Let's say the Cunningham chain starts with ap_n# + c. Then the next number in the chain is 2ap_n# + 2c + 1. c and 2c+1 are both prime since they are under primorial influence, so the corresponding Cunningham Chain starts with c.

Theorem

If there is a Cunningham Chain of the second kind where all values are under primorial influence and above the primorial, then there is also a corresponding Cunningham Chain of the second kind.

Let's say the Cunningham chain starts with ap_n# + c. Then the next number in the chain is 2ap_n# + 2c - 1. c and 2c-1 are both prime since they are under primorial influence, so the corresponding Cunningham Chain starts with c.

Theorem

If there is a Cunningham Chain of the first kind where all values are under primorial influence and below the primorial, then there is also a corresponding Cunningham Chain of the first kind.

Let's say the Cunningham chain starts with ap_n# - c. Then the next number in the chain is 2ap_n# - 2c + 1 = 2ap_n# - (2c - 1). c and 2c-1 are both prime since they are under primorial influence, so the corresponding Cunningham Chain starts with c.

Theorem

If there is a Cunningham Chain of the second kind where all values are under primorial influence and below the primorial, then there is also a corresponding Cunningham Chain of the first kind.

Let's say the Cunningham chain starts with ap_n# - c. Then the next number in the chain is 2ap_n# - 2c - 1 = 2ap_n# - (2c + 1). c and 2c+1 are both prime since they are under primorial influence, so the corresponding Cunningham Chain starts with c.

Consequences

I have used the principles above to search for Cunningham Chains near primorial numbers. Continue to the next page for information on the research.