HELP Mathematical notation needed for a prime mod 12 pattern

[JeremyEbert]

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

 

[JeremyEbert]

anyone?

 

[JeremyEbert]

should i post this in a different section?

 

[ramsey2879]

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

Since your looking for a response, I thought I point out that the equations
"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)"
Don't make sense to me and I feel that others may be having the same problem. Also the graphs add nothing tangible to me. But I am not savy in this matter anyway.
Edit you are in the right forum since the question deals in number theory. Do you have a proof for the statement that all Mersenne primes == 7 mod 12? It is obviously the case but can you prove it?

 

[JeremyEbert]

Thanks for the response! Sorry, I'm a coder that's why I need help.
Here is what I hope to be a better explination:

a + ((a-1)/2) ^2 = ((a+1)/2) ^2 ≡ b (mod 12)


if a =(6 * even) + 1 then a ≡ 1 (mod 12), b ≡ 1 (mod 12)

if a =(6 * odd) - 1 then a ≡ 5 (mod 12), b ≡ 9 (mod 12)

if a =(6 * odd) + 1 then a ≡ 7 (mod 12), b≡ 4 (mod 12)

if a =(6 * even) -1 then a ≡ 11 (mod 12), b≡ 0 (mod 12)



I figured there was already a proof for the Mersenne primes are ≡ 7 (mod 12). I'll work on that. What about all even powers of a prime are ≡ 1 (mod 12) and all odd powers of a prime are congruent to the base prime (mod 12)?

 

[JeremyEbert]

probably doesn't help but here is a visual representation of the mod 12 system with some notes.
http://1.bp.blogspot.com/_u6-6d4_gsSY/TDOJkPBDsVI/AAAAAAAAACE/d0emXHsXXYA/s1600/Prime+Equation2.gif

 

[ramsey2879]

Thanks for the response! Sorry, I'm a coder that's why I need help.
Here is what I hope to be a better explination:

a + ((a-1)/2) ^2 = ((a+1)/2) ^2 ≡ b (mod 12)


if a =(6 * even) + 1 then a ≡ 1 (mod 12), b ≡ 1 (mod 12)

if a =(6 * odd) - 1 then a ≡ 5 (mod 12), b ≡ 9 (mod 12)

if a =(6 * odd) + 1 then a ≡ 7 (mod 12), b≡ 4 (mod 12)

if a =(6 * even) -1 then a ≡ 11 (mod 12), b≡ 0 (mod 12)



I figured there was already a proof for the Mersenne primes are ≡ 7 (mod 12). I'll work on that. What about all even powers of a prime are ≡ 1 (mod 12) and all odd powers of a prime are congruent to the base prime (mod 12)?

Thanks for the explanation. Now it makes sense to me. As for the proofs, I am sure both have been proven but I think you should be able to come up with a proof. An induction proof will work for the Mersenne primes if you include 1 less than all odd powers of 2 above 1, not just 1 less than odd prime powers of 2. As to odd squares not divisible by 3, it is easy to show that they all = 1 both mod 4 and mod 3, what does that tell you about mod 12?

In response to your question I don't know what to call a residue system mod 12 other than just that.

 

[JeremyEbert]

Thanks for the explanation. Now it makes sense to me. As for the proofs, I am sure both have been proven but I think you should be able to come up with a proof. An induction proof will work for the Mersenne primes if you include 1 less than all odd powers of 2 above 1, not just 1 less than odd prime powers of 2. As to odd squares not divisible by 3, it is easy to show that they all = 1 both mod 4 and mod 3, what does that tell you about mod 12?

In response to your question I don't know what to call a residue system mod 12 other than just that.

Thanks again. There are features of this residue system mod 12 that I still need help with. I can depict them visually but I don't know the equations behind what I am envisioning. I know it starts with the "Inverse Square Law", mixes in some "Shell theorem" and "Pythagorean Triples" to show that a root of a number ultimately determines its primality.
If you look at the visualizations of all 3 equations it might help you understand my depictions a bit better.

Inverse Square Law:
http://upload.wikimedia.org/wikiped...uare_law.svg/500px-Inverse_square_law.svg.png

Shell theorem:
http://upload.wikimedia.org/wikipedia/commons/6/6f/Shell-diag-1-anim.gif

Pythagorean Triples scatter plot:
http://upload.wikimedia.org/wikipedia/commons/9/97/Pythagorean_triple_scatterplot2.png

My image 1: (notice the similar pattern as the Pythagorean Triples scatter plot)
http://2.bp.blogspot.com/_u6-6d4_gs...AAAE8/_hov_b0sno4/s1600/prime-+square+12a.png

My image 2: (notice the connection to the inverse square law and shell theorem)
http://4.bp.blogspot.com/_u6-6d4_gs.../bdPIJMIFTLE/s1600/prime-+square+12a+zoom.png

I hope this makes things clearer. I know its been shown that prime numbers and quantum phisics are closely linked via the Riemann Hypothesis. Should I post this in the physics section?

Thanks again for all your help.

edit:
This is the foundation of the images:
http://www.cs.cas.cz/portal/AlgoMat...onstructions/SquareSquareRootConstruction.htm

 

[ramsey2879]

Thanks again. There are features of this residue system mod 12 that I still need help with. I can depict them visually but I don't know the equations behind what I am envisioning. I know it starts with the "Inverse Square Law", mixes in some "Shell theorem" and "Pythagorean Triples" to show that a root of a number ultimately determines its primality.
If you look at the visualizations of all 3 equations it might help you understand my depictions a bit better.

Inverse Square Law:
http://upload.wikimedia.org/wikiped...uare_law.svg/500px-Inverse_square_law.svg.png

Shell theorem:
http://upload.wikimedia.org/wikipedia/commons/6/6f/Shell-diag-1-anim.gif

Pythagorean Triples scatter plot:
http://upload.wikimedia.org/wikipedia/commons/9/97/Pythagorean_triple_scatterplot2.png

My image 1: (notice the similar pattern as the Pythagorean Triples scatter plot)
http://2.bp.blogspot.com/_u6-6d4_gs...AAAE8/_hov_b0sno4/s1600/prime-+square+12a.png

My image 2: (notice the connection to the inverse square law and shell theorem)
http://4.bp.blogspot.com/_u6-6d4_gs.../bdPIJMIFTLE/s1600/prime-+square+12a+zoom.png

I hope this makes things clearer. I know its been shown that prime numbers and quantum phisics are closely linked via the Riemann Hypothesis. Should I post this in the physics section?

Thanks again for all your help.

edit:
This is the foundation of the images:
http://www.cs.cas.cz/portal/AlgoMat...onstructions/SquareSquareRootConstruction.htm

Sorry but I decline to go deeper into this than my understanding. My only advice is to first brush up on number theory and algebraic geometry. The graphs don't seem relevant to me and your presentation does not seem to fill in the gaps. It is hard to code such difficult topics if you can't put your thoughts into meaningful equations. Off the top of my head I would say that working in base 12 or mod 12 isn't the answer though.

 

[JeremyEbert]

Sorry but I decline to go deeper into this than my understanding. My only advice is to first brush up on number theory and algebraic geometry. The graphs don't seem relevant to me and your presentation does not seem to fill in the gaps. It is hard to code such difficult topics if you can't put your thoughts into meaningful equations. Off the top of my head I would say that working in base 12 or mod 12 isn't the answer though.

I know! That’s why I'm here. I am a very visual person unfortunately. I am much better on a white board than in a forum situation. It’s hard for me to get my point across without gestures and drawing. I guess that’s one of the many drawbacks of being self-taught. Thanks for looking. I am very aware that Base 12 only scratches the surface. I’m sorry I cannot present my ideas in meaningful equations. That ability is what I am seeking on this forum.

 

[JeremyEbert]

Does this help explain it better?

 

[JeremyEbert]

On the grid, whole number square roots are where the (x),(y) and a (circle) all intersect at once.

All other square roots (decimal numbers) intersect on a (circle) and the (x) according to their decimal value (y).

A composite number square root has more than one (circle),(x) intersection at its value (y).

A prime number square root only has one (circle),(x) intersection at its value (y).

This grid is just showing the divisibility of numbers and the fact that primes have no factor larger than one. Nothing surprising there. What I find surprising is that a numbers primality shows up at its square root. These prime roots only fall on the parabola in my graph that has a vertex of 1/2. All other numbers roots fall on multiple parabolas that have a vertex greater than 1/2 according to their factors. This graph is based on the Inverse Square Law which generally applies when some force, energy, or other conserved quantity is radiated outward radially from a point source. Its been shown that the non-trivial zeros of the Riemann zeta function have a real part equal to 1/2 and have a deep connection to the allowable energy levels in quantum systems that classically would be chaotic. I think this graph shows more of that connection.

 

[JeremyEbert]

no comments?

 

[TylerH]

You're looking at a specific case of a general pattern. I use it to cut down on the numbers I test for primality in my prime test.

What you're doing is finding all numbers coprime to 2 and 3. Effectively, it eliminates 2/3's of all numbers. The fact that a number is coprime doesn't guarantee true primality, though. Look at 49(49 mod 6 = 1), it passes, but is 7*7. The general case to find all numbers coprime to all primes less than or equal to p, where p is a prime:
n is coprime iff n mod p# = 1 or any prime between p and p#, where # denotes primorial.

In your case p is 3, p# is 6. 5 is the only prime between 3 and 6. So any number coprime to 2 and 3 will be congruent to 1 or 5 modulo 6.

I've explained before in this thread on a C programming board(so the language should be a little more native to you, being a programmer yourself) : http://cboard.cprogramming.com/c-programming/133651-how-use-large-numbers.html

 

[JeremyEbert]

You're looking at a specific case of a general pattern. I use it to cut down on the numbers I test for primality in my prime test.

What you're doing is finding all numbers coprime to 2 and 3. Effectively, it eliminates 2/3's of all numbers. The fact that a number is coprime doesn't guarantee true primality, though. Look at 49(49 mod 6 = 1), it passes, but is 7*7. The general case to find all numbers coprime to all primes less than or equal to p, where p is a prime:
n is coprime iff n mod p# = 1 or any prime between p and p#, where # denotes primorial.

In your case p is 3, p# is 6. 5 is the only prime between 3 and 6. So any number coprime to 2 and 3 will be congruent to 1 or 5 modulo 6.

I've explained before in this thread on a C programming board(so the language should be a little more native to you, being a programmer yourself) : http://cboard.cprogramming.com/c-programming/133651-how-use-large-numbers.html

Thanks for the reply! I have some very similar code. The elimination of 2/3's of all numbers goes a little further in mod 12 because you end up with 4 groups. Each group has only 1 mod12 congruence to check so you take 1/4 of the remaining 1/3 which gives you 1/12 of all numbers below the sqrt of the P to check.

My recent coding endeavors have been focused on how to exploit the pattern depicted in this PDF. I'd love to hear your thoughts.
https://www.physicsforums.com/attachment.php?attachmentid=31636&d=1296068205

 

[JeremyEbert]

update:
https://www.physicsforums.com/showthread.php?t=470799