Example of excluding primes from a proth-race candidate

Basically the powers of two, modulo, form a cycle. For some primes, e.g. 5, they take on every value (apart from 0), 1,2,4,3,1,2,4,3,... Therefore apart from when 5 divides k, k*2^n+1 (modulo 5) also takes on a loop of period 4 too. More interesting is 7, whose powers of 2 have shorter periods: 1,2,4,1,2,4,... Therefore (unless 7 divides k), k*2^n+1 (modulo 7) also has a period of 3. e.g. if k=10, k*2^n+1 (mod 7) will be 2,5,4,2,5,4,... and will never be zero, which is good - we've excluded 7 from being a prime factor of k*2^n+1, and thus boosted the prime density.

All the numbers that I was looking at had 7 as an excluded factor by construction, i.e were k == 0, 3, 5, 6 (mod 7). Also excluded are the following primes: 31 (period 5), 127 (period 7), 17 (period 8), 73 (period 9), 11 (period 10, can only be removed by putting it in the multiplier), 23 and 89 (period 11), and 13 (period 12 put in the multiplier).

This set of numbers which have all primes with Ord2(p)<=12 excluded is called the set E₊(12). They are all very good for generating large numbers of primes.

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