Observation on Primes under Primorial Influence

Observation

Primes that fall under the proven primorial influence that are members of a special class may have corresponding primes that are members of a special class.

From the previous page, primes in this range are defined by the primorial and a prime (or 1) for the equation:

a*p_n#+c

Primes where c = 1 are the special case called primorial primes and have been well studied. We are going to look, instead, when c is a prime.

An example:

A Twin Prime is a prime number that differs from another prime number by two. So the two numbers (11, 13) are twin primes. An example of twin primes under primorial influence are (41, 43) or (5#+11, 5#+13). Note that the two values added to the primorial are the example of the first pair of twin primes. Since 5#+11 and 5#+13 are twin primes under the primorial influence of 5#, then 11 and 13 must be twin primes. The converse is not true. The pair (7#+11,7#+13) are not twins since 7#+11 = 221 = 13x17 and is not prime.

These properties hold for all sequence within the proven primorial influence. Primes in this range that have special properties have the corresponding primes with the same special properties.

Consequence:

We can use this observation to search for primes of special forms that are near the primorials. For example, we could find all the twin primes between p_n and p_n+1 and then test each number a*p_n#+c where a=1,2,... and c is from the set of twin primes. By doing this, we have reduced the number of values to test and should be more likely to find twin primes.

Next, let's look at other special classes of primes.