Exploring Prime Numbers & Square Roots

[JeremyEbert]

I have read several books on the Riemann Hypothesis and have a general understanding of the non-trivial zeros and their real part 1/2. In my own studies I have devised a root system based upon some of Euclid’s ideas and congruence that identifies some interesting properties of the square roots of prime numbers. I have included a graph of the root system which I hope visually depicts some these properties and their uses in factoring. In my root system prime numbers only fall on the parabola with the vertex of 1/2. I wonder what relation, if any, can be compared to the non-trivial zeros and their real part 1/2?

http://4.bp.blogspot.com/_u6-6d4_gs.../bdPIJMIFTLE/s1600/prime-+square+12a+zoom.png

http://2.bp.blogspot.com/_u6-6d4_gs...AAAE8/_hov_b0sno4/s1600/prime-+square+12a.png

 

[JeremyEbert]

Interesting observance; 2 and 3 being the first two prime numbers make up the basic pattern in primes of 6(n)+-1 which accounts for 2/3 of all factorable numbers giving way to highly composite numbers. This factorability is the reason a base 12 system lends itself to grouping so nicely. With primes you get 4 groups of mod(p,12) outside of 2 and 3; = (1),(5),(7),(11). These 4 groups are derived from the pattern created by 2 and 3.
6(even) + 1 = (1)
6(odd) - 1 = (5)
6(odd) + 1 = (7)
6(even) – 1 = (11)
I like to refer to these groups as P1, P5, P7 and P11.
These 4 groups obviously contain composite numbers but they are nicely organized with their perfect square congruence. For instance, all P1 numbers have a mod (n1^2, 12) = 0 square congruence to mod (n2^2, 12) = 1 ( Group P1). The 4 groups’ square congruence is as follows;
P1 + (mod (n1^2, 12) = 0) == (mod (n2^2, 12) = 1)
P5 + (mod (n1^2, 12) = 4) == (mod (n2^2, 12) = 9)
P7 + (mod (n1^2, 12) = 9) == (mod (n2^2, 12) = 4)
P11 + (mod (n1^2, 12) = 1) == (mod (n2^2, 12) = 0)
Prime numbers: n1 = (P’ -1)/2 & n2 = (P’ +1)/2
Composite numbers: n1 <= (P’ -1)/2 & n2 <= (P’ +1)/2
Interesting results:
All Mersenne Primes are in P7.
All Prime Squares are in P1.
What would the proper equation for this be?