Observation on Primes under Primorial Influence

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Since a Cunningham Chains near primorials must have a corresponding Cunningham Chain we can use this property by choosing a small Cunningham Chain and searching for possible Cunningham Chains near primorials.

For example, we can look for Cunningham Chains of length 3 (CC3) near a multiple of 709# using the chain (719, 1439, 2879).  We will test (a709#+719, 2a709#+1439, 4a709#+2879) for a =1,2,...  After writing a program to do this, it discovered 484 probable CC3s within the first 100,000,000 values of a.  The program ran on a modest speed computer in less than eight hours to find these chains.  Optimizations to the program are in progress.  Below are some examples:

168996*709#+719  ..  675984*709#+2879
1349604*709#+719  ..  5398416*709#+2879
1448080*709#+719  ..  5792320*709#+2879
1500681*709#+719  ..  6002724*709#+2879
2322686*709#+719  ..  9290744*709#+2879

Because all values of the Cunningham Chains must be under primorial influence there are only ranges of numbers that can be used to find these chains with larger values.

For example a CC6 of the first kind can be searched for using:

a53# + 89  to  a83# + 89
a727# - 16651  to  a16649# - 16651

or for CC3 we can look in these ranges, for example:

a5# + 11  to  a7# + 11
a11# + 41  to a37# + 41
a17# + 89  to a83# + 89

to name a few.  Other ranges can be identified from the known Cunningham Chains.

Program Implementation

The program used to find probable primes uses the routines from the GMP library.  The first step is to set up a bit array for the values of 'a' so that they can be sieved.  Then, using test divisors, the remainder is found when dividing the primorial by that test divisor.  We find the first instance where aP#+c has a remainder of zero using the Extended Euclidian Algorithm.  We also test each of the numbers in the chain so that when we are done, there are no values in the test set that can be divided by small numbers.

The second step is to use the remaining values of 'a' and test them using the Miller-Rabin test routine as supplied in the GMP library.  We test each value in the chain until one fails or it is shown that all values are probable primes.

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Copyright 2008, Paul Schmidt, All Rights Reserved