Primes Near Primorials

Primes near Primorials

Are you interested in prime numbers?

This website discusses prime numbers that exist near primorials and how these numbers have some unique properties that make them interesting to study.

A primorial is the product of all prime integers up to a prime. Prime numbers near this primorial have some interesting properties that make them easier to find.

Primorial Notation:

p_n# is the primorial of p_n.

For example 7# is 210 (2*3*5*7).

A primorial prime is a number that is prime that is one more or less than a primorial. e.g. 13#-1 is a primorial prime.

It can be shown that numbers near (less than p_n+1² away from) a primorial, or multiple of a primorial, are only prime when it is a primorial prime, or the distance away from the primorial is prime, and that distance is greater than p_n. R. Potter has called the range of numbers near a primorial as being under Primorial Influence. For this website, we will only look at number that can be proven in this range (within p_n+1² of the primorial).

In mathematical notation:

a*p_n#+c is composite (not prime)

when:

c = 0
or c > 1 and c < p_n+1
or c > p_n+1 and c < p_n+1² and c is composite

Primes can only exist when c is 1 (primorial prime) or c is a prime above the primorial and less than the square of the next prime.

A program using this property was written to find primes within this range.

Consequences

Prime numbers under the primorial influence have the interesting property that they have a corresponding prime number that defines it. The consequence of this is that if these primes near a primorial are a special class of primes, the corresponding primes may also be of a special class. Some examples of special classes that are of interest are Twin Primes, Sophie Germain Primes, Cunningham Chains, and BiTwin Primes.

Continue on to find out more about these classes of primes and the implications.

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Consequences

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