Cracking The Prime Number Code

[Aethelwulf]

Can we find the pattern behind prime numbers?

A while ago now, I did uncover a relationship involving an approach to study the nature of prime numbers. Using the Vedic method of calculation showed me that possibly all prime numbers have a sum value that never comes to ''3'' apart from 3 itself - example:

99929

9+9+9+2+9 = 38

3+8 = 11

1+1 = 2

That is how you sum it. Another example

99719

9+9+7+1+9 = 35

3+5 = 8

The prime numbers sum value never comes to 3.... which is an interesting observation and believe it or not, actually hints at some underlying pattern. The tables of 3, 6 and 9 have interesting perturbative properties. A good example is

18 27 36 45 54 63 72 81

Do you see an underlying pattern here? the first four numbers of the nine times table (18 27 36 45) is an anagram of an ascending (12345678). So is the final four numbers up to 9 x 9, except the mirror symmetry moves down and up on opposite sides, i.e. 18 27 ascends the first two numbers, 72 81, is the final two and consists of rising and lowing numbers on opposite sides. What if the prime numbers appeared as an artifact of the perturbative properties of these related tables such that a pattern exists that can never be divided by any other number.

The best way to do this is to test it by analyzing the nine times table, and something amazing sprung out.

1-8 = 7
2-7 = 5
3-6 = 3
4-5 = 1
5-4 = 1
6-3 = 3
7-2 = 5
8-1 = 7

What do we have here? Notice, this is the nine times table, we have taken away instead of summing the values and what comes out? Prime numbers!

(2), 3, 5, 7, (11)

The interesting exception is in brackets (2) and (11) - notice, that in the table discovered above, 1 is indeed incorporated into the mix, and is not itself a prime numbers. It does occur twice and perhaps this accounts for the 2 perhaps, or the first emergence of 11?


A million dollar prize is fixed for anyone who finds the pattern behind prime numbers. I wish to apply this rule of thumb now to larger numbers associated to the nine times table and I hope to find the prime number pattern hidden throughout. I haven't done this yet, but I am doing so now.

If I am right, I'll collect my prize thank you

http://www.bbc.co.uk/news/magazine-14305667

http://www.physicsforums.com

[Aethelwulf]

Quickly studying higher numbers (excluding 90 and 99) as exceptions of being values 9 and can be excluded in hindsight that 18,27,36,45,54,63,72,81 (we exclude the 9 x 1 and 9 x 10, we have

 

 

108 = 8-1 = 7

117 = 7 - 1 - 1 = 5

126 = 6 - 2 - 1 = 3

135 = 5 - 3 - 1 = 1

144 = 4 - 4 - 1 = -1

153 = 5 - 3 - 1 = 1

162 = 6 - 2 - 1 = 3

171 = 7 - 1 - 1 = 5

180 = 8 - 1 = 7

 

 

Something odd happens at 144, it spits out something not contained in the pattern before. We may see something investigating more, maybe the same pattern will show up again? So far, what seems consistent is that two values of 1 keep appearing and so does 3, 5 and 7 the first three prime numbers (after 2) which never features in the tables created.

[Moontanman]

As usual dude you astound me, i hope you get the prize!

[Aethelwulf]

Thanks. I need to test it further... but it seems the base values springing out keep giving 7,5,3 and then 1. Then strangely ascends 1,3,5,7. The multiple 9 x 16 = 144 so far is an exception I need to investigate.

Edited February 11, 2013 by Aethelwulf

[Aethelwulf]

Sorry, in last post there I called 144 a prime number. This is of course not what I meant, that is 9 x 16.

[Aethelwulf]

Right... I have done this for more numbers... something strange happens. This is the previous set:

 

108 = 8-1 = 7

117 = 7 - 1 - 1 = 5

126 = 6 - 2 - 1 = 3

135 = 5 - 3 - 1 = 1

144 = 4 - 4 - 1 = -1

153 = 5 - 3 - 1 = 1

162 = 6 - 2 - 1 = 3

171 = 7 - 1 - 1 = 5

180 = 8 - 1 = 7

 

 

(Keep in mind, I have been taking the largest numbers intentionally to stay away from negatives. There might be something in the order of the operations that I have neglected. I will continue to ignore this) - now something happens when we continue this! 7 disappears from them, -1 now appears twice!

 

 

198 = 9 - 8 - 1 = 0 done in order (-6)

207 = 7 - 2 = 5

216 = 6 - 2 - 1 = 3

225 = 5 - 2 - 2 = 1

234 = 4 - 3 - 2 = -1

243 = 4 - 3 - 2 = -1

252 = 5 - 2 - 2 = 1

261 = 6 - 2 - 1 = 3

270 = 7 - 2 = 5

 

 

I will keep an update on more calculations I do. The more we do, maybe a bigger pattern will reveal itself, maybe a cyclic nature to the numbers? We have already arguably found a few.

[Aethelwulf]

Ok.... so I have done this for higher 9 times table numbers.

 

The pattern does break down eventually:

 

288 = -2

287 = -1

306 = 3

315 = 1

324 = -1

333 = -3

342 = -1

351 = 1

360 = 3

 

I guess it is to be expected when you get to larger numbers, negative numbers will be spit out. There has to be more to this, perhaps some kind of hidden pattern I need to look for. I don't believe for one moment the continued appearance of 7,5,3,1 is an accident for the first two cycles, but the appearance of negative -1 before did make it unusual. You do notice, that what made the pattern possible (the anagram of numbers 12345678) can be seen ascending and descending on both sides

 

18

27

36

45

54

63

72

81

 

on the right, it ascends, 12345678 while it descends on the left. You do see a residue of this pattern well into the 300's(x9) on the right hand side

 

288

287

306

315

324

333

342

351

 

What breaks the pattern perhaps, is that up the middle, the corresponding numbers to the right, don't begin to ascend until the forth number. On the left, we mostly have 3's. It is these patterns I am looking out for. I want to understand why these numbers appear.

Edited February 11, 2013 by Aethelwulf

[Aethelwulf]

378 = -2

387 = -2

396 = 0

405 = 1

414 = -1

423 = -1

432 = -1

441 = -1

450 = 1

459 = 0

 

Even to higher numbers, the pattern now seems... totally non-existent. A lot of -1's and -2, including frequent appearances of 1.

 

This seems just as random as the prime numbers! :P Neh, there is a pattern behind everything!

[Aethelwulf]

But there is one pattern I see:

 

A.

 

108 = 8-1 = 7

117 = 7 - 1 - 1 = 5

126 = 6 - 2 - 1 = 3

135 = 5 - 3 - 1 = 1

144 = 4 - 4 - 1 = -1

153 = 5 - 3 - 1 = 1

162 = 6 - 2 - 1 = 3

171 = 7 - 1 - 1 = 5

180 = 8 - 1 = 7

 

----------------------------------

 

B.

 

207 = 7 - 2 = 5

216 = 6 - 2 - 1 = 3

225 = 5 - 2 - 2 = 1

234 = 4 - 3 - 2 = -1

243 = 4 - 3 - 2 = -1

252 = 5 - 2 - 2 = 1

261 = 6 - 2 - 1 = 3

270 = 7 - 2 = 5

 

----------------------------------

 

C.

 

 

288 = -2

287 = -1

306 = 3

315 = 1

324 = -1

333 = -3

342 = -1

351 = 1

360 = 3

 

----------------------------------

 

D.

 

378 = -2

387 = -2

396 = 0

405 = 1

414 = -1

423 = -1

432 = -1

441 = -1

450 = 1

459 = 0

 

 

Notice in A., negative -1 appears once. In B., it appears twice. In C., it appears three times (but distributed). In D., it appears four times. It's frequency seems to increase by 1 in each evaluation of the numbers. Coincidence?

[Aethelwulf]

288 = -2

287 = -1

306 = 3

315 = 1

324 = -1

333 = -3

342 = -1

351 = 1

360 = 3

369 = 0

378 = -2

387 = -2

396 = 0

405 = 1

414 = -1

423 = -1

432 = -1

441 = -1

450 = 1

459 = 0

468 = -2

477 = -4

486 = -2

494 = 1

504 = 1

513 = 1

522 = 1

531 = 1

540 = 1

549 = 0

558 = -2

567 = -2

576 = -4

585 = -2

594 = 0

603 = 3

612 = 3

621 = 3

630 = 3

639 = 0

648 = -2

657 = -4

666 = -6

 

 

 

 

It struck me at this point, the pattern wasn't so much hidden in the result, but the pattern was hidden in the calculation itself.

 

It appeared that the numbers where cyclic because the results where coming from operations which where itself cycling. To explain what I mean, lets go back to the second:

 

 

108 = 8-1 = 7

117 = 7 - 1 - 1 = 5

126 = 6 - 2 - 1 = 3

135 = 5 - 3 - 1 = 1

144 = 4 - 4 - 1 = -1

153 = 5 - 3 - 1 = 1

162 = 6 - 2 - 1 = 3

171 = 7 - 1 - 1 = 5

180 = 8 - 1 = 7

 

Look at this part

 

A - B - C

.........

7 - 1 - 1

6 - 2 - 1

5 - 3 - 1

4 - 4 - 1

5 - 3 - 1

6 - 2 - 1

7 - 1 - 1

8 - 1

 

I've made this into three files A - B - C.

 

Even though, the order of these simple calculations are made to keep away from negative numbers, we later find out we cannot escape them indefinately. When you take the order of the calculations I have done, you can see up the files, some interesting cyclic property of the nine times table. On the left A-file, you can see 7654567, in the B file, 1234321... and the C file reserved for the repeated numbers more than twice. Certain patterns will appear if these files (with the patterns involved) are constantly shifting but still converging enough to make some patterns. In theory, I'd guess, this pattern would go on for ever.

 

Having a look at my latest numbers

 

459 = 9 - 5 - 4

468 = 8 - 6 - 4

477 = 7 - 7 - 4

486 = 8 - 6 - 4

495 = 9 - 5 - 4

504 = 5 - 4

513 = 5 - 3 - 1

522 = 5 - 2 - 2

531 = 5 - 3 - 1

540 = 5 - 4

 

And looking at the same property, we do have some ascending and descending numbers.

 

Focusing on the second file, we have 5676543234 - one can see properties clearly. I guess, in theory, if I calculated enough of these, the numbers will eventually shift back to 1 corresponding to 2 ect. Complete guess. Also, it may not actually have anything to do with prime numbers, I am only basing this investigation on the fact that three of the first set of prime numbers appeared from the early calculations. But if there was any pattern in the prime numbers, it seems logical at least to think that such a pattern can be investigated from the well known primes, which is a sense of irony since we are investigating primes in this thread from well known divisible examples.

[Aethelwulf]

Prime numbers themselves, when looked at visually do represent a kind of pattern:

 

 

http://en.wikipedia.org/wiki/Ulam_spiral

 

 

If there is a real pattern there, I bet my money that it is a result of cyclic properties of numbers like we have investigated so far.

[Turtle]

Can we find the pattern behind prime numbers?

 

no.

 

A while ago now, I did uncover a relationship involving an approach to study the nature of prime numbers. Using the Vedic method of calculation showed me that possibly all prime numbers have a sum value that never comes to ''3'' apart from 3 itself - example:

 

99929

 

9+9+9+2+9 = 38

 

3+8 = 11

 

1+1 = 2

 

That is how you sum it. Another example

 

99719

 

9+9+7+1+9 = 35

 

3+5 = 8

 

The prime numbers sum value never comes to 3.... which is an interesting observation and believe it or not, actually hints at some underlying pattern. The tables of 3, 6 and 9 have interesting perturbative properties. A good example is

 

18 27 36 45 54 63 72 81 ...

 

the process is called taking the digital root [in base ten per your example; the process works in any base]. in base ten all natural numbers with digital roots of 3, 6, & 9 divide by 3 and therefore are not prime.

 

 

Prime numbers themselves, when looked at visually do represent a kind of pattern:

 

 

http://en.wikipedia.org/wiki/Ulam_spiral

 

 

If there is a real pattern there, I bet my money that it is a result of cyclic properties of numbers like we have investigated so far.

 

over the past several years here at Hypography, i and a few others have explored Ulam's spiral as well as a number of other number spirals of our own invention/design in the thread Non-figurate Numbers. you may want to review it so you don't have to repeat our work. given your interest in math here, i'm surprised you have overlooked it.

[Aethelwulf]

over the past several years here at Hypography, i and a few others have explored Ulam's spiral as well as a number of other number spirals of our own invention/design in the thread Non-figurate Numbers. you may want to review it so you don't have to repeat our work. given your interest in math here, i'm surprised you have overlooked it.

 

 

Thank you very much, I have never came across this thread in all the time I have been here! Hard to imagine... but true. I shall read it now.

Thanks!

[CraigD]

Turtle beat me to the post, denying me the prize of first to mention digital roots.  I have a good excuse, though – I woke up late, and ran into a work commitment before I could finish my post.

 

Though I’m repeating some of what Turtle said, here’s my belated reply:

 

A while ago now, I did uncover a relationship involving an approach to study the nature of prime numbers. Using the Vedic method of calculation showed me that possibly all prime numbers have a sum value that never comes to ''3'' apart from 3 itself - example:

That’s a cool observation, Aetherwulf! :thumbs_up

 

I checked it for the first 10,000,000 primes, and it’s true. It’s also true for 6 and 9 – none of the first 10,000,000 primes had digital roots (that’s the usual term for the iterative summing of digits until a single digit number is found).

 

As best I know, this coincidence has been known for a long time. It’s mentioned in the wikipedia article I linked, which I think (didn’t read the whole article, but it seemed to be going there) sourced it from a 1961 New Scientist magazine article.

 

I’ve never seen a proof of it, but think it would be a pretty easy one. I can’t offer a $1,000,000 hidden prize like Marcus du Sautoy, but anyone posting one would get my “knows how to write a mathematical proof” nod of approval for posting one, and the satisfaction of having, if even in a small way, engaged in math in public. ;)

 

A million dollar prize is fixed for anyone who finds the pattern behind prime numbers. I wish to apply this rule of thumb now to larger numbers associated to the nine times table and I hope to find the prime number pattern hidden throughout. I haven't done this yet, but I am doing so now.

 

If I am right, I'll collect my prize thank you ;)

The link in the BBC article was broken, but the 2011 pages are archived here at good old archive.org. Nobody will send you the big prize Aethelwulf – you’ve got to do all sort of video sleuthing and puzzle solving (some of it, presumably, involving prime factorization) to get scavenger hunt clues that’ll lead you to some sort of physical location (presumably in England) where the $$s are hidden.

 

The really big prize – as in you get to be known as the person that made all RSA-type dual key encryption, and thus the underlying cryptography relied on by 99%+ of all public data communication, insecure, is for coming up with a computationally efficient (ie: computing time proportional to log(number to be factored) or less) algorithm for factoring the product of 2 large “strong” (ie: not guessable via special case factoring tricks) primes, like the 617 decimal digit (more commonly described by the number of binary digits in its prime factors, 1024 bits) number in the BBC article.

 

Some of my first hypography posts, such as this 2005 one, were aimed toward this prize. I, and a hoard of folk both more and less able than I, am still hunting. :) I’m guessing you’re in this hunt too, Aetherwulf, recently or for a while.

[Turtle]

Turtle beat me to the post, denying me the prize of first to mention digital roots.  I have a good excuse, though – I woke up late, and ran into a work commitment before I could finish my post.

 

Though I’m repeating some of what Turtle said, here’s my belated reply:

 

 

That’s a cool observation, Aetherwulf! :thumbs_up

 

I checked it for the first 10,000,000 primes, and it’s true. It’s also true for 6 and 9 – none of the first 10,000,000 primes had digital roots (that’s the usual term for the iterative summing of digits until a single digit number is found).

 

As best I know, this coincidence has been known for a long time. It’s mentioned in the wikipedia article I linked, which I think (didn’t read the whole article, but it seemed to be going there) sourced it from a 1961 New Scientist magazine article.

 

I’ve never seen a proof of it, but think it would be a pretty easy one. I can’t offer a $1,000,000 hidden prize like Marcus du Sautoy, but anyone posting one would get my “knows how to write a mathematical proof” nod of approval for posting one, and the satisfaction of having, if even in a small way, engaged in math in public. ;)

...snip

 

:lol: a coup!? maybe just counting coup. a rare event & honor at any rate to best you at anything. :D

 

anyway, we both linked to the same article & i believe they give a proof for base 10 [i,e, mod 9] as well as for the general case of any base B_n. for my own example, any natural number written in base 9 divides evenly by 8 if it has a digital root of 8, and by extension that number divides by 4 & 2. so 56₁₀=62₉ dr(62₉=6+2=8

 

here's wiki's proof: (still noodling it myself...:smart:)

Digital root @ wiki

Proof that a constant value exists

 

How do we know that the sequence eventually becomes an one digit number? Here's a proof:

 

Let , for all , is an integer greater than or equal to 0 and less than 10. Then, . This means that , unless , in which case n is an one digit number. Thus, repeatedly using the function would cause n to decrease by at least 1, until it becomes an one digit number, at which point it will stay constant, as .

 

Congruence formula

 

The formula is:

or,

To generalize the concept of digital roots to other bases b, one can simply change the 9 in the formula to b - 1.

 

The digital root is the value modulo 9 because and thus so regardless of position, the value mod 9 is the same – – which is why digits can be meaningfully added. Concretely, for a three-digit number,

 

To obtain the modular value with respect to other numbers n, one can take weighted sums, where the weight on the kth digit corresponds to the value of modulo n, or analogously for for different bases. This is simplest for 2, 5, and 10, where higher digits vanish (since 2 and 5 divide 10), which corresponds to the familiar fact that the divisibility of a decimal number with respect to 2, 5, and 10 can be checked by the last digit (even numbers end in 0, 2, 4, 6, or 8).

...

 

they also -interestingly- link from that page to an article on the vedic square, something i never saw referenced 'til today. :omg: it seems the first posting at wiki was ~2008. anyway, the vedic square is my old "katabatak" multiplication table base 10 that i put up here in '05. of course, when you change bases as i do in the katabataks, there are an infinite number of such tables. curioser & curioser.

 

attaching my base 9 [mod 8] "vedic square" for perusal and per se elucidation.

[Turtle]

Thank you very much, I have never came across this thread in all the time I have been here! Hard to imagine... but true. I shall read it now.

Thanks!

 

you're welcome. :) i'd start at the end and read back as it's rather a lengthy thread. :omg: there is however something of a summary of results at the end of post #1, though i'm not sure i have updated that with any reference to the spiral arrays. looking forward to any insights you may get from -or have on- it all. :smart:

[Turtle]

Prime numbers themselves, when looked at visually do represent a kind of pattern:

 

 

http://en.wikipedia.org/wiki/Ulam_spiral

 

 

If there is a real pattern there, I bet my money that it is a result of cyclic properties of numbers like we have investigated so far.

 

returning to the op and primes, i'm attaching my original "turtle" spirals, one of equilateral triangles and the other of hexagons. these are the only other regular polygons that tile the plane as do the squares of Ulam and as far as i have found i am their original inventor/discoverer. (i implore anyone who knows otherwise to notify me here with a reference. ) primes are red cells in the equilateral-triangle array; you can safely ignore the rest of the colors here. black cells on hexagon array are prime. as with the ulam spiral, rows of numbers have quadratic expressions on these arrays. :smart: