Cracking The Prime Number Code

[G Anthony Kent]

The human propensity to see patterns in ironic occurences is constanly amazing. Your prime numbers games are nothing new. Such games have been played for centuries. What would be more interesting is to find a connection between the iterative Koch equation for the prime numbers and the iterative relations defining fractal chaos patterns having complex variables. One could write the Koch equation in the complex plane and then try to relate it to some sort of fractal chaos function. The result might be to point the way to cracking public encryption codes with arbitrary numbers of digits in their prime number bases.

 

 

see the Koch equation in the attachment

Edited March 2, 2013 by Gak

[Turtle]

The human propensity to see patterns in ironic occurences is constanly amazing. Your prime numbers games are nothing new. Such games have been played for centuries. What would be more interesting is ...

 

First, you extoll probing primes, then you minimalize the activity, then you suggest you have something to contribute but seem to imply your approach is no game. Covering all the bases are you? :doh: Perhaps then you are the one to find a connection between the iterative Koch equation for the prime numbers and the iterative relations defining fractal chaos patterns having complex variables. Talk is cheap in any age. :blahblahblah:

[CraigD]

Talk is cheap in any age. :blahblahblah:

True, but the prime number generator Gak showed,

see the Koch equation in the attachment

though computationally intense beyond practical utility, is cool! :thumbs_up It’s prime-determining term,

[math]\prod_{j=0}^{t-2} \prod_{k=0}^{t-2} (jk-t)[/math]

is intriguing – that Koch was a pretty deep fellow!

 

Where did you find this, Gak? I’ve not seen it before, even in the best, broadest, and most mathematically intense book I know on the subject, John Debyshire’s Prime Obsession.

 

Due to its computational extravagance, I could only actually evaluate the first formula for n=1 and 2, but it seems to work. The second “Kentgen's modification of” formula doesn’t, on inspection, because it’s powers or 2 terms are always even.

[Aethelwulf]

Oh there is definitely something behind this digital root system stuff. I've been racking my brains over it for a while.

 

Something to do with the properties of the appearance of the prime numbers coupled with them not converging on the 3, 6 or 9 digital root system, as Craig has mentioned. The fact they don't converge is evidence there is a pattern.

Edited March 3, 2013 by CraigD
Merged 2 short posts

[Turtle]

Oh there is definitely something behind this digital root system stuff. I've been racking my brains over it for a while.

 

Something to do with the properties of the appearance of the prime numbers coupled with them not converging on the 3, 6 or 9 digital root system, as Craig has mentioned. The fact they don't converge is evidence there is a pattern.

 

i strongly disagree. not about brain wracking as that's a given, but that there is any master formula employing digital roots that produces only primes. (sieve of Eratosthenes excepted.) the non-convergence is simply numbers that divide by 3 and so by definition are not prime. Eratosthenes 2^nd iteration.

 

Talk is cheap in any age. :blahblahblah:

True, but the prime number generator Gak showed,

 

though computationally intense beyond practical utility, is cool! :thumbs_up It’s prime-determining term,

[math]\prod_{j=0}^{t-2} \prod_{k=0}^{t-2} (jk-t)[/math]

is intriguing – that Koch was a pretty deep fellow!

 

i'm glad you made something of it because i was like.... :blink: better to do useless things than do nothing at all i suppose. nevertheless and cool notwithstanding, Gak associated the useless equation with a fractal, and as a fractal is a visual "thing", i damn well want to see one. :rant: :lol: can you make one Craig?

[CraigD]

It’s prime-determining term,

[math]\prod_{j=0}^{t-2} \prod_{k=0}^{t-2} (jk-t)[/math]

is intriguing – that Koch was a pretty deep fellow!

Alas, I believe I was dull-minded from staying up late when I wrote this, and distracted by the typographical challenge of parsing the equations.

 

There’s nothing especially deep about this expression. It’s simply the equivalent of a program that tests the primality of [imath]t[/imath] by comparing the product of every pair of natural numbers less than it, [imath]jk[/imath], to it, by subtraction.

 

What would be more interesting is to find a connection between the iterative Koch equation for the prime numbers and the iterative relations defining fractal chaos patterns having complex variables.

That would be interesting, but I’ve seen no evidence that a profound such connection exists.

 

As I've explained above, Koch's prime-generating equation above is essentially a simple, inefficient non-prodcedural program. I can see no connection between such a program and a complex number-using (or equivalently, 2-dimensional) fractal shape generating program such a the Mandelbrot set. Koch's famous fractal snowflake is not generated in this way, so I can find no connection there. Koch made an important proof concerning prime numbers, but that proof appears to me unrelated to fractal shape generators like the Mandelbrot set or the Koch snowflake.

[Aethelwulf]

i strongly disagree. not about brain wracking as that's a given, but that there is any master formula employing digital roots that produces only primes. (sieve of Eratosthenes excepted.) the non-convergence is simply numbers that divide by 3 and so by definition are not prime. Eratosthenes 2^nd iteration.

 

 

 

 

 

Let's not forget, the digital root system is not a division process. That doesn't define a prime like you are saying above. Prime numbers cannot be divided evenly by any other number. The digital root system is different - we are in effect analyzing the appearance of digits using a vedic calculation. If anything has a pattern, it will be found here because, apart from 3, 6 9 all other numbers appear in the vedic summation of digits in prime numbers.

 

Why?

 

I can only conclude there is a pattern we are currently not seeing.

Edited March 4, 2013 by Aethelwulf

[Turtle]

Let's not forget, the digital root system is not a division process.

 

yes; it is equivalent to modulo division. in base ten, 786=7+8+6=21=2+1=3 & 786 mod 9 = 3 the remainder after dividing 786 by 9, is 3.

 

change base and the divisor of the modulo function is 1 less than the base. so, for example, the digital root in base 8 is mod 7. 786 mod 7 = 2. write 786 in base 8 = 1422=1+4+2+2=11=1+1=2

 

That doesn't define a prime like you are saying above. Prime numbers cannot be divided evenly by any other number. The digital root system is different - we are in effect analyzing the appearance of digits using a vedic calculation. If anything has a pattern, it will be found here because, apart from 3, 6 9 all other numbers appear in the vedic summation of digits in prime numbers.

 

Why?

 

I can only conclude there is a pattern we are currently not seeing.

 

write yourself a list of primes in base 10 and take the integer root of each, minding it's equivalent to finding the remainder of each prime after dividing by 9. yes the results are all 1 2 4 5 7 or 8. [with the one-time exception of the prime 3] nonetheless, there is no consistent order to these roots/remainders. well, insofar as i have looked but i have looked far enough to see if such a pattern existed, it would be too large to see.

 

it is fun i admit. if you like calculations, and i know you do, you could write out the integer roots of the primes for base 3, base 4, base 5, ... equivalent to finding Prime_x mod 2, mod 3, mod 4 ...

 

{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97 ...}

integer roots of primes base ten: {2 3 5 7 2 4 8 1 5 2 4 1 5 7 2 8 5 7 4 8 1 7 2 8 7 ...}

[Turtle]

... snip

 

write yourself a list of primes in base 10 and take the integer root of each, minding it's equivalent to finding the remainder of each prime after dividing by 9. yes the results are all 1 2 4 5 7 or 8. [with the one-time exception of the prime 3] nonetheless, there is no consistent order to these roots/remainders. well, insofar as i have looked but i have looked far enough to see if such a pattern existed, it would be too large to see.

 

it is fun i admit. if you like calculations, and i know you do, you could write out the integer roots of the primes for base 3, base 4, base 5, ... equivalent to finding Prime_x mod 2, mod 3, mod 4 ...

 

{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97 ...}

integer roots of primes base ten: {2 3 5 7 2 4 8 1 5 2 4 1 5 7 2 8 5 7 4 8 1 7 2 8 7 ...}

 

as luck has it, courtesy his majesty modest, i/we have actual look-atable tables for mod3 through mod399 from another pursuit. :omg: so for example i have attached modest's mod189 table which i modified to show only primes.

 

NOTE: this table is the equivalent of writing primes in base 190 notation and taking the digital roots of each. now, if you want to be true to how we use base notation, you need 190 unique characters. supposing we have those characters, though we really have at easy hand 62 [arabic numerals, upper sase letters, lower-case letters.] but, a base 189 number would have at least those and might use only them. so what is the value of 2Zg₁₈₉? let alone the digital root 2+Z+g=?? you see the problem? :wacko:

 

this attached table and the others start from 1 in the upper left corner and moves right the number of cells coincident with the modulo divisor, and then the numbering restarts in column 1, row 2 and goes right, etc. . so, any cell in column 10 has a remainder of 10 when divided by 189, etc. . the table height(s)is 400 cells. so the table below is 189x400=75,600 natural numbers. primes are colored blue; not-primes black.

 

so, see anything pattern-wise here that repeats over/within the interval? if not, anysuch pattern if it exists would have to be at least twice as big, which is to say the pattern of the first 75,600 naturals would have to repeat for the next 75,600. :omg:

 

let me know if you want to see a particular diagram. :coffee_n_pc:

Edited March 4, 2013 by Turtle

[Aethelwulf]

I had a dream last night... about the prime numbers. I was holding two lines papers, one shifted down, the other shifted up, matching numbers. Then there was a second, a square diagram which matched others numbers up.

 

I woke up thinking, that prime numbers can be solved in a geometric way.

Edited March 4, 2013 by Aethelwulf

[CraigD]

I woke up thinking, that prime numbers can be solved in a geometric way.

In waking life terms :), what do you mean by “prime numbers can be solved”, Aethelwulf?

 

To me, “solving the prime numbers” means “being able to find the prime factors of any integer n in time (by which I mean number of elementary arithmetic operations, eg: adding or multiplying single numerals – see below why I’m taking pains to state this) on the order of log n – that is, in an amount of time proportional to the amount of time it takes to simply read or copy n. A further condition is that we can represent n in a positional numeric way (eg: as a string of binary or decimal digits). Let’s call this the fast factoring problem, or FFP.

 

I think it’s useful to outline what prime-related math things we already can, and can’t do:

Here some things we can do:

• Generate a dense (no missing elements) list of the prime numbers 2 through the mth prime in time on the order of m log m (eg: the
  
• Generate a non-dense list of primes through w in time on the order of log w
  
• Calculate the nth prime P_n in time on the order of ... the best I can find is “very inefficient” and “contrived the the extent they are of little practical value”.
  

These abilities are of little value for the FFP.

 

Your dream, Aethelwulf, appears to be of a way of generating a dense list of the first m primes in time on the order of m, or finding the nth prime in time order log(n). These, too, would be of little value for the FFP.

 

Some more things we can do:

• Probabilistically determine, with arbitrary certainty, if a number n is prime or composite, in time on the order of log n (example: the Miller-Rabin test)
  
• Determine with certainty if n is prime in time on the order (log n)^m, where m>1
  

In practical effect, this means that, for any number we can represent as given above, we can determine if it is prime or not. What they cannot do, in general, is determine a prime factor of the number if it is not prime.

 

In short, what the FFP needs is not a way of finding a particular prime given the prime that precedes or follows it, or given its position in the list of all primes, but a way of finding a prime given a composite number containing it. The preceding sentence is just a longer phrase for my initial paragraph’s “find the prime factors”, and the even shorter verb “factor”.

 

Though my personal list of prime related math things we can’t do is longer, and the world’s much longer, for this post, it’s short:

• The FFP
  

Notice that I was careful to define “time” (sometimes called “time complexity”) as a count or arithmetic operations, not the clock-measured duration of a person or machine performing them. Otherwise, the FFP could be taken out of the realm of mathematics into that of physics by solving it with a large quantum computer, which could (if they are actually physically possible, which I believe is an open question) essentially make an arbitrarily large number of guesses to find a factor of any composite number in a short amount of time. In short, no cheating by tapping the practically infinite multiverse via quantum computers! ;)

[tetrahedron]

Hey Turtle, regarding your hexagonal array above- the Machine Consortium wants their multiple dimensional page back, otherwise they are gonna send Jodie Foster to deal with ya.

[Turtle]

Hey Turtle, regarding your hexagonal array above- the Machine Consortium wants their multiple dimensional page back, otherwise they are gonna send Jodie Foster to deal with ya.

 

The Machine Consortium can suck my Unity and kiss my Galactic center. :omg: I hold the Galactic copyright & I will defend it to the last iota. Of course, Jody is welcome anytime. :kiss:

 

Here's another page from my Prime catalog. Each arm is Prime_x units long.

 

[Turtle]

Hey Turtle, regarding your hexagonal array above- the Machine Consortium wants their multiple dimensional page back, otherwise they are gonna send Jodie Foster to deal with ya.

 

me again. :doh: so regarding my hexagon array of primes above... again...- the Modest Collaborateur last april produced a computer generated hexagon array of some 250000+ cells. following my plan, modest color keyed the original of that image for polygonal numbers and 7 classes of non-polygonal numbers. the image post is here: >> modest's post #1131, Non-figurate Numbers. the code may be somewhere a few posts either side of it.

 

So....Primes being one of those classes, i have now recolored all non-primes black & left the Primes Red. attached is scaled to 50% of original, but i think sufficient detail is preserved. talk about crackin' some code! props & genuflections in modest's directions y'all.

 

Edit: add link to modest's image.

Edited March 6, 2013 by Turtle

[Turtle]

following up, naturally, modest also made me a large computer-generated special-keyed [equilateral]-triangle spiral array. [modest's post & code here, or within 6 posts either side as we tweeked it: >> post #1065, Non-figurate Numbers ]. given the inherent difficulties of mapping equilateral triangles on a square matrix, the triangles aren't equilateral in the image and the hexagon boundary is irregular. nevertheless, the relative positions are correct. i don't know how many cells in this one off-hand; a butt-load i'll wager. :omg: anyway, i recolored Non-Primes Black & Primes Red as for the hexagon array. i didn't scale this one down as the assymetry makes for some squirching i didn't care for.

 

i want to give a shout-out and a gigundous prop & genuflection in Tormod's direction for giving me special dispensation to post numerous large images lo these many years. :thumbs_up you da man T!!!

 

enjoy!

 

Edit: add link to modest's code

 

some modest comments on the Primes:

Good deal :)

 

 

... Because the size of the rings grows predictably (and the size of the sides of the ring in particular) it was easy to say that any 6n²+1 hexagonal ring would have all of its odd numbered triangles point left... so all the primes should be on those and point left as well. Probably a lot of the other classes will follow suite

 

 

Speaking of primes, one thing I found very interesting and unexpected is the great number of primes arranged on some of the lines. For example,

 

 

if I counted right this line (and in particular, the left-pointing triangles on it) takes on the form 6n²+17. The density of primes on 6n²+17 appears to be quite large — 40 out of 60 on the section I just outlined. :blink:

 

 

...

~modest

Edited March 6, 2013 by Turtle

[Turtle]

Let's not forget, the digital root system is not a division process. That doesn't define a prime like you are saying above. Prime numbers cannot be divided evenly by any other number. The digital root system is different - we are in effect analyzing the appearance of digits using a vedic calculation. If anything has a pattern, it will be found here because, apart from 3, 6 9 all other numbers appear in the vedic summation of digits in prime numbers.

 

Why?

 

I can only conclude there is a pattern we are currently not seeing.

 

 

i trust you read my explanation on this earlier? good. :thumbs_up when i first started with residue multiplication i was working algebraically; a nephew suggested putting the results in the table. bright boy. :smart: from that launch i constructed so called "vedic tables" for bases 2 through 15. note the vedic table in base 2 is a single square/cell. a 1x1 table. anyway, i went to check for them in the katabatak thread but they are gone from the posts, causalties of my gallery disappearance after a forum software change. who ya gonna call?

 

so i'm attaching the base 15 table to illustrate what i earlier said about digital roots & division. because this is truly written in base 15, we need characters for 10, 11, 12, 13, & 14 .following the standard these are A B C D & E respectively in/on the table. [Key:1=Black, 2=Red, 3=Orange, 4=Yellow, 5=Green, 6=Blue, 7=Indigo, 8=Violet, 9=White, 10=Charcoal, 11=Brick=Red, 12=Brick-orange, 13=Brick-yellow, & 14=Brick-green]

 

now, in base 15 the integer root of numbers that divide by 14 [one less than the base as i said] is E. moreover, in base 15, numbers that divide evenly by 7 have digital roots of 7 if they are odd. more moreover, in base 15 any number with an even digital root is even. {2 4 6 8 A C E}, so the only digital roots primes can have in base 15 are {1 3 5 9 B & D}

 

 

so to some primes and their digital roots base 15; the first 25.

 

 

Primes (base 10) {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97 }

Primes (base 15 ) {2 3 5 7 B D 12 14 18 1E 21 27 2B 2D 32 38 3E 41 47 4B 4D 54 58 5E 67 ...}

Digital roots of primes base 15 {2 3 5 7 B D 3 5 9 1 3 9 D 1 5 B 5 B 1 3 9 D 5 D ...}

 

so the remainder when you divide 97 by 14 [97mod 14] = 13 & 13 in base 15 notation is D. clear as mud yet?

 

 

 

 

EDiT: add Primes in Base 15

Edited March 16, 2013 by Turtle

[Turtle]

i trust you read my explanation on this earlier? good. :thumbs_up when i first started with residue multiplication i was working algebraically; a nephew suggested putting the results in the table. bright boy. :smart: from that launch i constructed so called "vedic tables" for bases 2 through 15. note the vedic table in base 2 is a single square/cell. a 1x1 table. anyway, i went to check for them in the katabatak thread but they are gone from the posts, causalties of my gallery disappearance after a forum software change. who ya gonna call?

 

... snip

 

 

aha!! while the tables are gone from the thread, and the current gallery is glitchy, i found all my katabatak tables/vedic squares for bases 2 through 16. this link should be a whole folder and the multiplication tables are mixed in. looks like i numbered these over the original coloring code.>> :artgallery: Studio K