Adventures In PRIME NUMBER Land !!! ... (A Dummies Guide to Prime Numbers)

Wow, you've really discovered something here!

You're partially right on the reason why large primes are so valuable for data encryption methods. Factorization of two multiplied primes is very difficult indeed -- but also, the fact that a prime is divisible only by one and itself is of great value to prevent partial brute force hacks.

Depending on the encryption method (I'll use the old LanMan or LM hash as an example) to conserve space the key may actually be stored (or can be hacked) in chunks that are factors of the original key length.

In the LM hash (used to encrypt passwords in every Windows OS prior to Vista) your 14-character password is actually stored in two 7-character chunks. So, you really only need to hack two 7-character passwords rather than one 14-character password.

It sounds like the same amount of work (hacking hacking two 7-char strings rather than one 14-char string) but it's really a *LOT* less work. To show this, I'll give some examples of how many permutations exist for each 7-char chunk of the password, versus a single 14-char password.

To make this simple I'm going to use every lowercase letter of the alphabet as a possible character in a password. Yes I realize you can also use numbers or symbols (which is a good practice by the way) but for simplicity's sake we're going to stick with lowercase letters.

So, if your password can be at most 14 characters long, and you have 26 available options for each character (one for each letter in the alphabet) the number of unique possible combinations are:

26! 26! 26x24x23x22x21x20x19x18x17x16x15x14x13x12x11...x1
26_P_14 = ------- = ---- = -------------------------------------------------------------------
(26 - 14)! 12! 12x11...x1

Basically, you subtract the length of the string (14) from the number of available characters (26) giving you 12. Next, you take the factorial of the number of available characters (26) down to 12. Factorials are easy. The factorial of 26 = 26 x 25 x 24 x 23 x 22 x 21.... etc. all the way down to 1. So, we take the factorial of 26 down to 12:

26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 = 10,103,301,395,066,880,000

That makes for roughly 10 QUINTILLION possible combinations for a password 14 characters long. Now let's do the same for a 7-character password, which means we take the factorial down to 19:

26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 = 62,990,928,000

As you can see, this is a SIGNIFICANTLY smaller number of possible combinations. Over 100 million times as small. In the world of brute-force hacking this is a tremendous advantage. For those of you unfamiliar with the term, "brute force" hacking means basically trying every available combination of characters until you happen to get it right. If I know your username, I just "brute force" the server or website you logon to, trying every available combination of letters until I happen to get it right.

Let's say my computer is super dooper fast and can brute-force 1,000,000 passwords per second. Using the figures above, for a 7-character password, it would take about 62,990 seconds -- or about 17 hours.

Using the same computer, to brute-force a 14-character password would take 10,103,301,395,066 seconds, or roughly 320,000 years, give or take a few. Obviously, hacking two 7-char passwords would be a better choice.

(Continued from first post)

I'm surprised no one has mentioned any possible implications here . . .

Based on your diagram and formulas, this could potentially greatly reduce the amount of time required for de-factorization of multiplied primes used in encryption technologies. The elusive "n" would certainly make it REALLY easy but even without, the fact that you can quickly reduce your number of possible combinations down to those on the prime rays is of great value.

Your formula could quickly determine whether an exhaustive test would be necessary on any candidate number in the defactorization process.

I wish I wasn't inundated with work right now I'd love your help to write a program, using this formula, to find some of this "monster" primes with rewards offered by EFF and Prime95

Originally posted by tauristercus
So what we end up with is an infinite number of concentric circles, each divided up into 24 segments

You have not plotted an infinite number of circles. You plotted 10. I want to see the findings repeated flawlessly in 100 circles to give your statement any credibility. This is rather like early mathematicians computing pie to 10 places and claiming that is all.

A star and a flag for you my friend. Enough said.

The fact that all star numbers appear to be located on rays 1 and 13 is really interesting too.

Assuming this holds true out to infinity, finding new star numbers (and star primes) could be alot easier as well, especially given that star numbers ALWAYS end in 01, 13, 21, 33, 37, 41, 53, 61, 73, 81, or 93.

Just find a number ending in one of the above, on Ray 1 or Ray 13, darn good candidate for a star prime.

Primes are obviously a base-10 phenomena and I'm not even sure if there's a "prime number equivalent" in any other base ... now you've got me thinking ... well done !


[

No. Primes are primes regardless of the number base.

Many other possible patterns arise when the primes are represented in different number bases.

I have often thought that we may be missing something about prime numbers simply because of our obsession with base-10 OR even our accepted numeral system for that matter

Simply amazing thread. Starred.

This kind of understanding of primes could undoubtably be used to futher the efficiency of computer programs I bet. I can see a couple already that may help with some 3d games

Originally posted by way2slo
1. 41 x 41 = 1681
2. 1681 / 24 = 70.0416666666666 (repeating)
3. 70 + 1 = 72
4. 0.0416666666666 x 24 = 1

Really?

edit:
Nevermind looks like someone already covered that. But awesome post I will have to come back to this again!

[edit on 20-9-2009 by way2slo]

I know, I know ... make one little typo and everyone catches it !

But in a way that's good because it means you're taking the time and trouble to follow my reasoning ... well done, guys

Originally posted by Mortimer452
(Continued from first post)

I'm surprised no one has mentioned any possible implications here . . .

Based on your diagram and formulas, this could potentially greatly reduce the amount of time required for de-factorization of multiplied primes used in encryption technologies. The elusive "n" would certainly make it REALLY easy but even without, the fact that you can quickly reduce your number of possible combinations down to those on the prime rays is of great value.

Your formula could quickly determine whether an exhaustive test would be necessary on any candidate number in the defactorization process.

I'm really pleased you picked up on the possibility that there may be a way to greatly reduce the number of primes needed to be checked for factorization. I actually did mention something along those lines that squaring the product of the two original primes essentially immediately eliminates EVERY prime on the other 7 rays (rays 2 thru 8) requiring only to search along that single Ray1.
Also, a further bit of assistance is that an upper limit (a ceiling, so to speak) on primes needed to be searched along Ray1 would be provided by the squared value of the product of the two primes used. Only primes less than this value would need to be evaluated.

And if we could come up with some way of "plugging" the required value into n, well, many encryption schemes based on primes would take a beating !

Originally posted by YourForever

Originally posted by tauristercus
So what we end up with is an infinite number of concentric circles, each divided up into 24 segments

You have not plotted an infinite number of circles. You plotted 10. I want to see the findings repeated flawlessly in 100 circles to give your statement any credibility. This is rather like early mathematicians computing pie to 10 places and claiming that is all.

I totally agree with you regarding the point you've raised ... in my defense I have used many more circles and all primes have continued to fall on only those 8 rays ... so up to that point, nothing "breaks".
The good thing is that anyone who cares to can go right ahead and plot as many circles as they feel like doing and seeing if they can disprove what I'm stating.

Also, instead of drawing 100's of circles, another way of indirect proof is by selecting as many primes as you care to consisting of a small, medium, large and very large number of digits and use the examples I provided to see if they consistantly fall on ONLY those 8 rays.
If only one prime can be found that DOES NOT fall on any of those 8 rays, then I'll be the 1st to admit my thread and assumptions were wrong

In fact, I would welcome such tests by ATS members

[edit on 21-9-2009 by tauristercus]

Your work on this has me amazed. I recently started coming to ATS to read the topics on the front page that interested me but yours is the first that has actually gotten me to finally create an account and post.

I sat here for the past hour trying to find an equation that would satisfy n. Obviously an hour is nowhere near enough time to figure it out because you have probably been working on it wayyyy longer than that. But I cant help but feel that there is a relationship between n and the star numbers. I spent most of that hour trying to find a relationship between n and star numbers working from the bottom up with your example with 11 * 127.

Your work really has my attention even though I am not the greatest at math!

Originally posted by sylintcc08
Your work on this has me amazed. I recently started coming to ATS to read the topics on the front page that interested me but yours is the first that has actually gotten me to finally create an account and post.

I sat here for the past hour trying to find an equation that would satisfy n. Obviously an hour is nowhere near enough time to figure it out because you have probably been working on it wayyyy longer than that. But I cant help but feel that there is a relationship between n and the star numbers. I spent most of that hour trying to find a relationship between n and star numbers working from the bottom up with your example with 11 * 127.

Your work really has my attention even though I am not the greatest at math!

This sort of response (yours) makes the effort I put into creating this thread well worthwhile as I have to admit I was initially concerned that because there was no "inbuilt conspiracy" or "imminent catastrophy in the making", that few people would be interested in my ramblings ... but what a surprise to find this level of interest !

Also, I was hoping to get other points of view that might have a substantial impact on what I've been showing and perhaps others might "see" things that I've missed ... this Star Number idea that MickC, Mortimer and yourself have mentioned, being an obvious example !

And yes, the quest for the elusive n replacement method is definitely still under way

Just out of curiosity...how long have you worked on this?

I am truly impressed with your work. S&F for sure once I get off my mobile!

A2D
(this thread and others like it are a prime[no pun intended] example of why ATS is so amazing)


Posted Via ATS Mobile: m.abovetopsecret.com

Originally posted by Agree2Disagree
Just out of curiosity...how long have you worked on this?

I am truly impressed with your work. S&F for sure once I get off my mobile!

A2D
(this thread and others like it are a prime[no pun intended] example of why ATS is so amazing)

 

Posted Via ATS Mobile: m.abovetopsecret.com

 

Thanks for the encouragement !

Admittedly, this hasn't been an overnight project for me as I've been messing around with it (on and off) for about 2 years now.
Based on some of the responses and ideas posted by others so far in this thread, looks like it'll be occupying my attention for quite some time yet to come as I feel there's still lots of surprises waiting to be revealed about the primes

Reply to post by tauristercus

I am glad to see there are still those with an unwavering dedication. I wish you the best of luck with your adventures in primes, and life in general. I fear I can do very little assistance-wise for lack of mathematical prowess. I suppose it's satisfying enough to have atleast partaken in this knowledge.

Thank you again! Thanks as well to all those mathematically inclined that help in this process. Maybe someday my grandchildren will learn about primes through your methods.

A2D


Posted Via ATS Mobile: m.abovetopsecret.com

Your diagram is very simple to explain. I will explain and provide a proof that there are infinite number of primes laying on one of your prime rays and never on the other rays.

For ray #1, the numbers are 24n + 1. No factors.
For ray #2, the numbers are 24n + 2. Factors are 2 and 12n + 1. Hence, the numbers laying on ray #2 will always be composite.

In short,

24n + 1 (No factors)
24n + 2 = 2*(12n + 1)
24n + 3 = 3*(8n + 1)
24n + 4 = 4*(6n + 1)
24n + 5 (No factors)

etc etc.

So rest assured, your diagram is correct.

I find the rest of your post fascinating. I have been studying prime numbers for a long time. I will study your posts for further analysis.

reply to post by Deaf Alien

So rest assured, your diagram is correct.

Deaf Alien, thanks for providing that additional confirmatory proof ... hopefully it satisfies any potential doubters regarding the validity of the claim that primes will ONLY be found on rays 1 thru 8.

reply to post by tauristercus


Another interesting find

I was analyzing your claim that every prime number squared will always lay on Ray #1. Upon brief analysis, it appear that you are correct. I haven't complete the analysis.

For all prime numbers in ray #1 (actually any number laying there):

(24n + 1)^2 = 24^2 * n^2 + 2 * 24*n + 1 = 24*(24*n^2 + 2*n) + 1

Which is 24*k + 1 which proves that it lays on ray #1

For all prime numbers in ray #5:

(24n + 5)^2 = 24^2 * n^2 + 2 * 5 * 24 * n + 25 = 24^2 * n^2 + 2 * 5 * 24 * n + 24 + 1 = 24(24*n^2 + 10n + 1) + 1

Which is 24*k + 1 again.

I have not worked out the rest yet. I'm getting tired LOL.

Outstanding thread

Have you heard of cymatics?

Look at the following pics and see how the observed nodes of resonance match your rays, albeit with a factor divisible by 3, 6, 12, 24!

First one with six observable nodes:
[atsimg]http://files.abovetopsecret.com/images/member/c37278405a43.jpg[/atsimg]

Second with twelve observable nodes:
[atsimg]http://files.abovetopsecret.com/images/member/93367f4423df.jpg[/atsimg]

And finally, a video showing how sound generates nodal patterns and i suspect when the note reaches some division of your formula (regarding rays and primes squared), then the sand sets in a pattern, where the peaks and troughs force the sand into place.



And the pièce de résistance, is how i vehemently believe that by resonating magnetism at certain frequencies, it will be possible to create a UFO or anti-gravity machine as it resonates with the ethereal 'galactic current' of electro-magnetism that permeates the Multiverse.

It seems prime numbers and the Golden Ratio are both ubiquitous in nature and tantalisingly close in our scientific understanding to be able to harness ZPE, if indeed the g'ments haven't already known about this technology and suppressed the knowledge.

Here's a basic picture i drew, of a device i'd love to build with neodymium magnets spinning on the outside of a core of mercury plasma toroid

[atsimg]http://files.abovetopsecret.com/images/member/06c26fdb1616.png[/atsimg]

Finding patterns is always very interesting.

I was looking at your concentric circles and the I saw a tunnel. Basically each circle is a set of 24 numbers. So primes appear on the rays you designated. There are also non primes on these rays. So the rays indicate a set of 8 numbers of which some are prime and some are non prime. Just eyeballing and counting I see 4, 5, or 6 (maybe 7 or 8 in the first circle) primes in each set of 8. If 1 in prime and 0 in no prime you get something like the following.

1=11111111 --1, --5, --7, -11, -13,- 17, -19 ,-23
2=01101111 ---, -29, -31, ---, -37, -41, -43, -47
3=01011100 ---, -53, ---, -59, -61, ----, -67, -71
4=10111100 -73, ---, -79, -83, ---, --89, ---, ---
5=11111100 -97, 101,103,107,109,--113, ---, ---
6=01111100

and so on. Don't know how accurate the numbers are above it's late. I just wonder if something like this may help identify a pattern.

So right now your rays are right about half the time or better in predicting primes. In each column (or ray) the numbers are in next line are either 24, 48 or 72 more until the next prime. Don't know how long this holds true.

So keep up the good work.

[edit on 21-9-2009 by A52FWY]

[edit on 21-9-2009 by A52FWY]

[edit on 21-9-2009 by A52FWY]