The sequence of Prime numbers can be defined using the fact that Primes are one away from a multiple of a=1, 2, 3, 4, 6. In general, we can say that a Prime, other than 2 and 3 some times, is of the form p= a*k(a)±1 for some k(a) <> a*x*y ± x ± y for any (x,y) positive integer.

The generation of primes using this algorithm is complete based on the following observation:

1: With k=6xy+x+y, we have 6k+1 = 36xy+6x+6y+1 = (6x+1)(6y+1), i.e. all products of two factors both equivalent to +1 (mod 6);

2: With k=6xy-x-y, we have 6k+1 = 36xy-6x-6y+1 = (6x-1)(6y-1), i.e. all products of two factors both equivalent to -1 (mod 6);

3: With k=6xy-x+y, we have 6k-1 = 36xy-6x+6y-1 = (6x+1)(6y-1), i.e. all products of two factors, one equivalent to +1 (mod 6) and the other equivalent to -1 (mod 6).

Starting with the integers equivalent to ±1 (mod 6) and excluding these three sets leaves those integers equivalent to ±1 (mod 6) which cannot be represented as a product of two factors equivalent to ±1 (mod 6), i.e. the primes p≥5.

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