- #1
JeremyEbert
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OK, I need help putting this into mathematical notation.
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.
The pattern is defined the best when you deal with square roots and ultimately I need the mathematical notation to represent these visualizations I have attached. They are detailed images. Please download them and zoom in if needed.
http://4.bp.blogspot.com/_u6-6d4_gs.../bdPIJMIFTLE/s1600/prime-+square+12a+zoom.png
and
http://2.bp.blogspot.com/_u6-6d4_gs...AAAE8/_hov_b0sno4/s1600/prime-+square+12a.png
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.
The pattern is defined the best when you deal with square roots and ultimately I need the mathematical notation to represent these visualizations I have attached. They are detailed images. Please download them and zoom in if needed.
http://4.bp.blogspot.com/_u6-6d4_gs.../bdPIJMIFTLE/s1600/prime-+square+12a+zoom.png
and
http://2.bp.blogspot.com/_u6-6d4_gs...AAAE8/_hov_b0sno4/s1600/prime-+square+12a.png