Need help with prime Test

*To*: mathgroup at smc.vnet.net*Subject*: [mg124905] Need help with prime Test*From*: KenR <ramsey2879 at msn.com>*Date*: Sat, 11 Feb 2012 06:39:41 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com

I need to generate more "Ramsey" Numbers to further verify that only prime numbers can meet the criteria. Ramsey numbers are those generated from a simple criteria that is easy to check. I imported a CSV list into my Mathematica version 8 to check all 30,759 numbers that my Excel macro selected and they all turned out to be prime. That is all that my program selected from all odd numbers from 3 to 1,048,655, as large as my Excel program could test. I would further test my program using my Mathematica software but I would like some help to write the most efficient program to do the job. Right now, I am trying to write a while loop inside a do loop but don't know how to exit the loop before the counter becomes 0 in the case that it is clear that P is not prime. The check is to do the following binary recursive sequence mod P and check to see that the (P-1)/2 term is zero and no term prior to that is zero. If so then I believe P should be prime based upon the results so far. The test sequence is S(0) = 2, S(1) = 3, S(n) = 6*S(n-1) - S(n-2) - 6. It appears that S((P-1)/2) is divisible by P then P is very likely Prime, but I am interested in a test that is valid only for primes. S((35-1)/2) is divisible by 35 but that is not the first term divisible by 35. Only about 1/3 of the primes seem to meet the more restricted criteria, i.e. the sequence has no term divisible by P prior to S((P-1)/2).