The role of p squared

p squared:  4,9,25,49,121….represents the stepping stones of prime number distribution: the pattern changes at each step




p#: 2, 6, 30, 210, 2310……represents the length of the pattern at each step, hidden by the next step up, so that the full pattern never shows itself, other than at the beginning.

{Π(pn – 1) } / pn # or 1,2,8,48,480,5760….. over p#

This is the proportion of primes in each repeatable element derived from Eratosthenes sieve

Thus prime number distribution between 4 and 9 is remarkably simple – every other integer is prime! 5,7; from 9 to 25 life becomes more complex – the interaction of prime factors 2 and 3 lead to the famous pair pattern followed by a gap: 11,13…17,19….23…(and then a new pattern starts at 25 complicating matters further: the pattern between 25 and 49, which is 30 integers long, only starts to repeat itself at 55, buried under a new pattern starting at 49)

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