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DNA Primes Math Blog

DNA Primes Math BlogDNA Primes Math BlogDNA Primes Math Blog
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  • Prime Numbers
  • Riemann Zeta
  • Infinite Sums/Products
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DNA PRIME THEORY

PRIME NUMBER STRUCTURES

PRIME NUMBER STRUCTURES

PRIME NUMBER STRUCTURES

The Structure of Prime Numbers

  

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.

PRIME NUMBER STRUCTURES

PRIME NUMBER STRUCTURES

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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