Ramsey Primes are those generated from a simple criteria that is easy to check. I checked all odd numbers from 1 to 1 million and 29455 numbers met the criteria. None were composite. The check is to do the following sequence mod P and check to see that the (P-1)/2 term is zero and no term prior to that is zero. The test sequence is S(0) = 2, S(1) = 3, S(n) = 6*S(n-1) - S(n-2) - 6. If P is prime, then S((P-1)/2) is divisible by P, 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 primes seem to meet the more restricted criteria, i.e. have no term divisible by P prior to S((P-1)/2). The first two Ramsey primes are 11 and 13 which I call a Ramsey Twin. The largest Ramsey Twin under 1 million is (998651, 998653). Any thoughts are welcome. Edit So far about 5 in 13 primes are Ramsey Primes so my test has limited value unless it can be proven that only primes can meet the test. Any way to prove this?