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

reply to post by pavil


I would love to hear how he explains how he sees numbers and arithmetics but alas I couldn't find any video with subtitles. I've been watching this google video The Boy With The Incredible Brain. I could get some gist of what was going on.

I would love to see his drawings of numbers but I've only found a couple. That will help us understand the numbers, especially prime numbers, tremendously.

I an normally one of those people who would say, "Where's the conspiracy?" But NOT for YOUR POST!!!!!

Billiant. This should be in a math journal! I have a masters degree in electrical engineering and am very into heavy math in the process of doing research and engineering for the defense industry. For me, my opinion is that you have stubbled upon something larger than we even know yet. There is something more to be explored here. I don't know what it is yet, but it is important! I'm really excited about this article. I'm going to chew on it for a while and see if I can come up with any ideas.

Keep up the good work!

Whether discovered or rediscovered (after reading some replies) I have to say that your post was educational to say the least. The fact that I learned something, found value in it and was able to teach the idea to others in a casual conversation makes this a very valuable thread.

Yeah, hi...

it seems to be loosely based on "9"
i think it occurs because you will always reach 99 eventually, and then 99 leads to 297 which locks it in at that value

very cool pattern u found


reply to post by GORGANTHIUM

In my previous post yesterday, I showed how a few very simple arithmetic steps could be used to produce an infinity of odd numbers.

In this post, I've decided to go the visual route using geometry, as an alternative way of visualizing prime number creation. Often, it's easier to "see" something rather than having it described in words or maths.

Take a look at the following diagram and you'll see how easy it is to use geometry to generate primes.

Simply start with a single "unit" which gives us our starting prime of 1.
Then to generate the next prime (5), just add 24 more "units" to create a square totaling 25 units ... and of course, the square root of 25 is 5, another prime.

To get the next prime (7), just add another 24 "units" giving 49 units in total ... square root of 49 is 7, another prime.

In fact, as many primes as required can be generated simply by adding as many units of 24 as may be required, and taking the square root when required.

I suppose you could look at this as being the "geometrical equivalent" of Ray 1 that contains the squares of every prime number along its length.

Anyway, the diagram is basically self-explanatory ... but the main point I want to stress yet again, as I've been doing thru all my posts and examples, is how the value 24 seems to come up over and over and over when we're dealing with prime numbers.

[atsimg]http://files.abovetopsecret.com/images/member/a99f4fa245f5.jpg[/atsimg]

Sorry I don't really think you've discovered anything new here. What you are really doing is eliminating the non-prime numbers; this can not be used as a method of finding primes.

Note that the rays on which the primes occur must also be numbered with a number which is relatively prime to 24. To see why, please consider the following argument.

For ray number r, all th enumbers along the ray are of the form:

r+24n

Where n is the number of the circle.

Note that no even numbered ray can contain primes, since then r=2*k and we have that all the number along the ray are of the form 2k+24n= 2(k+12n) and hence are divisible by two.

If you follow this reasoning, you can convince yourself that only the rays who's number is relatively prime to 24 can contain any prime numbers since all of the numbers on other rays must be divisible by a common factor of the ray number r and 24.

Consider the ray 9. However, 9 and 24 have the common factor 3. Therefore, all the numbers on ray 9 can be written as 3*3+3*8*n=3(3+8n) and therefore all of these numbers are divisible by 3. So no numbers along this ray can be prime.

The problem is, as you extend your rays that do contain primes out to infinity, they will also contain other non-prime numbers. You see this already in your small example but the primes get less dense at higher values and you will see this as you extend your idea. As a result, you still have to test each number along the ray to see if it is prime and most of them will not be. This is decreasing density of the prime number distribution is known as the prime number theorem.

FWIW, yes I have a degree in mathematics and I have studied number theory although this is not my specialty.

See also:

mathworld.wolfram.com...
en.wikipedia.org...
www.ulamspiral.com...

Upon further review, I see that someone else already pointed this out to you. For some interesting new research on prime number patterns, check out this:

www.physorg.com...

1. First of, there is nothing new or discovery like in what the OP has posted. (Or perhaps only here in ATS).

2. What is ACTUALLY going on, is another way to display the already known relationship of prime numbers. The relationship is known how they are scattered. What the OP did, was that he represented the prime numbers in base-24. There's no need to use the base-24 symbolic system to achieve this since primes always fall on the same slots. Well, you can use e.g. base-12 to represent primes and what you get is equally nice (visually).

Try this:

Write the colums of 12 numbers (numbers are in base-10):


01 02 03 04 05 06 07 08 09 10 11 12
13 14 15 16 17 18 19 20 21 22 23 24
25 26 27 28 29 30 31 32 33 34 35 36
37 38 39 40 41 etc.


Now colour the prime numbers and can you see the pattern? Same idea but different presentation than in the original post.

You can do this with any base-system. Eg. base-12

x = Not prime
P = Prime


x x P x P x P x x x P x
P x x x P x P x x x P x
x x x x P x P x x x x x


3. What is MORE important is that you cannot predict (correctly) the next prime number with this system. There is no known formula for the prime numbers. Again, what was done was purely representational "trick".

4. Also, you cannot say if number X is a prime number EVEN if it falls in the right cluster.

reply to post by tauristercus


this... this was actually one of the more fascinating things I have read on this site or elsewhere in quite a while. It gave me a lot to think about on the subject of number theory. It would see that given this, there would be a potentially novel way to use this to scan for primes using some more efficient programming algorithms if there haven't been some made already (the sieve method comes to mind for spirals). moreso, it would be easy to restart at an arbitrary number, and could 'find' its ring and ray relatively trivially. main thing would be to make an abstract class to handle ginormous numbers.

ok, programmer side of me coming out and obviously brimming with possibilities.

but anyways, awesome work, i'd star it 2x if i could

reply to post by tauristercus



Great post! There definately seems to be relationship between prime numbers, the square of 12(144), the number 24, the golden mean and ratio, Fibbonacci numbers, fractals, the Pythagorean sacred numbers, sacred geometry etc.. Possibly the secrets of antigravity, thus tapping into the power of hyperdimensional physics. Check out this video on the secrets of Coral Castle, Fla.:

Coral Castle

But I'm afraid the ancients have one upped you on this. They seemed to have working knowledge of many things that we are just starting to try to understand today.

Also, check out the most famous fractal set, besides the Fibbonacci sequence, which is the Mandelbrot set: www.youtube.com...

It appears that Einstein was right, and that the order of the universe seems to come down to a single mathemetical, fractal set, and it all seems based on the primes numbers, and the numbers 24 and 144,




[edit on 22-9-2009 by HothSnake]

[edit on 22-9-2009 by HothSnake]

[edit on 22-9-2009 by HothSnake]

Lets use the following prime numbers to illustrate this effect ... 41, 1009 and 10007.

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

Therefore, 1681 (41 squared) is on Circle 72, Ray1

Your math in step three is wrong, 1681 would be located on circle 71.

Also, while I understand your reasons for accepting the number 1 as a prime; and I can understand omitting the number 2 for its being even,
I failed to see anywhere a reason why the number 3 should be neglected other than it simply didn't fit with your system. I don't think its proper to put much faith into this while the number 3, which meets the requirements for primality, isn't included.

Other than these, I found this topic to be incredibly interested. A few months ago I fell into this yearning to learn about the primes, most of my research was conducted with Ulam's Rose. This is the most interesting writing concerning primes that I have read.

Originally posted by dangrsmind
Sorry I don't really think you've discovered anything new here. What you are really doing is eliminating the non-prime numbers; this can not be used as a method of finding primes.


FWIW, yes I have a degree in mathematics and I have studied number theory although this is not my specialty.

If the elimination of non-prime numbers the way it was presented in its essence cannot find primes, would you be able to demonstrate the method which does the job in its numerical, not abstract/generalized form?

The OP mentioned all the arguments that you presented and concluded with prime factorization, the applicable part of primes. He just used a long lemma to get there as an intro for those who are not overly familiar with the subject.

[edit on 9/22/2009 by stander]

Originally posted by tauristercus

In this post, I've decided to go the visual route using geometry, as an alternative way of visualizing prime number creation. Often, it's easier to "see" something rather than having it described in words or maths.

A visual renditions are the coolest and quickest way to deliver the essence of an argument. Unfortunately, mathematicians must use proof-ready language when presenting their ideas, but this esoteric language is not understood outside the domain of expertise. And so many relatively simple ideas are obscured by these requirements.

The prime circle of yours is fun, even though it is not an original insight. That takes you back to the ancient times when numerology and number theory lived under one roof. The ancient number crunchers were discovering patterns and other relationships in their study of quantities, but the time they lived allowed them to make various attributes to the origin. Even millenia later Einstein remarked upon the way God made the universe. And so the ancients were discovering patterns that worked until they didn't -- an unexpected number showed up and the miracle was gone. The people tend to remember these spoilers and that's one reason for the existence of bad and good numbers.

Here is an example: Start with axiom 1 + 1 = 2. Now re-arrange the expression to look like this: 11 + 2 =. That prompts you to complete the addition. If you do, you get a subsequent prime to 11. Now repeat the process of the addition (recurrence) by adding the next even number to the result. You are going to see primes . . .

11 + 2 = 13 (prime)
13 + 4 = 17 (prime)
17 + 6 = 23 (prime)
23 + 8 = 31 (prime)
31 + 10 = 41 (prime)
41 + 12 = 53 (prime)
53 + 14 = 67 (prime)
67 + 16 = 83 (prime)
83 + 18 =101 (prime)

That's quite unexpected run for the ancient number cruncher who was sort of aware of the unpredictable incidence of primes and of the absence of a simple formula that would generate them. Is some far reaching discovery in the making?

But the series is due to a chance. The spoiler is knocking on the door . . .

101 + 20 = 121 (composite)

I tested this on a few professional mathematicians, and guess what. All of them failed to spot the relationship between the first and the last terms.

first: (11) + (2) = [13]
last: [101] + [20] = (121)

The ( ) relationship is easy: 11^2 = 121.

What kind of relationship applies to [ ] so 101, 20, and 13 would form a simple true expression similar to 11^2=121?

You have a better chance to find it than a pro mathematician. The reason why these guys failed to see the relationship is that the rendition is not analytic in nature. Their mind have acquired heavy stereotypes and doesn't allow them to venture beyond the analytic paradigm. But that's normal -- nothing wrong with that.

Being interested in math and unrealized potential to come up with some order to chaos I have enjoyed this thread a lot. I have also enjoyed venturing out for more information on the subject brought up by the talented OP as well as others. I do however have to call the OP out again and direct the thread to the work of Dr. Peter Plichta and His Book God's Secret Formula. It would seem from my research that he is the originator of the idea of the prime number cross and 24.



And as stated by others in this thread its not really that revolutionary if you understand that
All primes greater than 3 are of the form 6k-1 or 6k+1
Start Ulam's Sprial at 41 and see this...



That said it is brilliantly presented and is very interesting. It the distribution of the primes or understanding how many more 24's have to be added to find the next prime number that is really what is being sought and I'm not sure these ideas will get us there. Look at the links below the first is in german but g oogle will translate enough to get a gist.

www.plichta.de...

Site discussing the Dr

Fun stuff, I hope to see this thread keep going and get more from the OP!

A very good explanation of what is going on... Interesting read

8.3 Ulam Rose? Get a Grip!
www.abarim-publications.com...

Originally posted by WoodEye

And as stated by others in this thread its not really that revolutionary if you understand that
All primes greater than 3 are of the form 6k-1 or 6k+1

It would be fun if this property was indigenous only to the primes. Unfortunately it's not so.

There was this ancient guy who was presented with the above in the form of an equation to solve:

(primes +- 1) / kx = a remainder 0, where k=1,2,3,4,5...

That dude used the long-forgotten "outside the box" analytic method: he counted the number of letters in the parameter primes and that was it.
PRIMES = 6 = x.

Now what was the name of that person?

Pimesprimesprimes?

Actually, I think that somewhere near the infinity, there is a prime, which when added or subtracted by 1 and then divided by 6, doesn't return a whole number. Can anyone show me that such a prime doesn't exist without making the trip to the infinity?


[edit on 9/22/2009 by stander]

reply to post by stander

Actually, I think that somewhere near the infinity, there is a prime, which when added or subtracted by 1 and then divided by 6, doesn't return a whole number. Can anyone show me that such a prime doesn't exist without making the trip to the infinity?

" ... near the infinity ... " ?

Not sure what you mean by that statement as there is no such thing as being "near" infinity.

As there are an infinity of primes, you can select ANY prime that you want (no matter how big) and from that point on, you can be guaranteed that there will still ALWAYS be an infinity of primes after it.

So, based on the above, I'm not really sure what you're getting at.

Originally posted by tauristercus
reply to post by stander

 

Actually, I think that somewhere near the infinity, there is a prime, which when added or subtracted by 1 and then divided by 6, doesn't return a whole number. Can anyone show me that such a prime doesn't exist without making the trip to the infinity?

" ... near the infinity ... " ?

Not sure what you mean by that statement as there is no such thing as being "near" infinity.

As there are an infinity of primes, you can select ANY prime that you want (no matter how big) and from that point on, you can be guaranteed that there will still ALWAYS be an infinity of primes after it.

So, based on the above, I'm not really sure what you're getting at.

That's strange that you couldn't grasp the meaning of "near infinity," when you use the term "infinity of primes," which is very misleading, coz infinity is not a number, nor is it a synonym to "bunch of" or "plenty of" primes.

Here is the semi-symbolic equivalent to what I meant if it helps you to understand my drift.

There is a prime number p>10^1000000000000000000000000000000000 where mod(p +- 1 , 6) = q and q 0

Someone mentioned that if you take any prime larger than 6, either subtract or add 1 and then divide the result by 6, you get always an integer. Now consider a series of subsequent whole numbers a(1), a(2), a(3) ... a(n-1), a(n) where the last term is aproaching infinity (a(n) --> infinity symbol). That a(n) happens to be a prime number that doesn't satisfy the property mentioned above. Is there any way to show that it does without finding the exact numeric value of the prime, which is impossible anyway, coz a(n) is approching infinity and its value cannot be therefore determined? In other words, is there a proof lingering around the statement that for ALL primes (prime +- 1)/6 = a, remainder 0?

I think the proof is trivial, but I can't figure it out.

reply to post by stander

That's strange that you couldn't grasp the meaning of "near infinity," when you use the term "infinity of primes," which is very misleading, coz infinity is not a number, nor is it a synonym to "bunch of" or "plenty of" primes.

Nothing strange about it as in maths, the term "infinity", usually denotes an unbounded limit.
In this thread, we are concerned only with the set of positive integers n, where n grows without bound, i.e. Ip = [1,2,3,4,5, ...., n].

By saying "near infinity", one can then reasonably ask "just how close is that really to infinity ... are we talking withing spitting distance ?", as if the term "infinity" itself represented some kind of super-duper humongous number.
But no matter how close you may like to think you've chosen a number close to "near infinity", there will always be an infinity more of numbers between the number you picked and .... well, infinty, I guess :-)

Don't make the common mistake of thinking that the concept of "infinity" can be equivalent to a number, because it can't.
That's like asking what happens when I divide 10 by infinity, or multiply 5 by infinity ... can't do it because infinity is NOT a number ... therefore NOT being a number, you can't get "near to it".


Anyway, let's not worry too much about semantics ... ok ?

Let me see if I can answer your question:

Here we'll use modular arithmetic in base 6.
If we assume that 2 and 3 are primes, then it naturally follows that all multiples of 2 and 3 greater than 2 and 3 are not primes.

In mod 6, there are six subdivisions of the positive integers (n)
0 mod 6 = 6n and giving answers such as 6, 12, 18, 24, 30, ...
1 mod 6 = 6n + 1 and giving answers such as 1, 7, 13, 19, 25, ...
2 mod 6 = 6n + 2 and giving answers such as : 2, 8, 14, 20, 26, ...
3 mod 6 = 6n + 3 and giving answers such as : 3, 9, 15, 21, 27, ...
4 mod 6 = 6n + 4 and giving answers such as :4, 10, 16, 22, 28,...
5 mod 6 = 6n + 5 and giving answers such as : 5, 11, 17, 23, 29, ... (Note that 6n+5 is equivalent to 6n-1)

Now, any positive integer that is equivalent to 0 mod 6, 2 mod 6, or 4 mod 6 must be even and therefore will be divisible by 2.
e.g 6n/2 = 3n, (6n + 2) / 2 = 3n + 1, (6n + 4) / 2 = 3n + 2.
This means that any such positive integer can't be prime unless it's actually the number 2.

Also, any positive integer that is equivalent to 3 mod 6 is divisible by 3.
e.g. (6n + 3) / 3 = 2n + 1.
This means that any such positive integer can't be prime unless it's actually the number 3.

So after having eliminated the above, this leaves us with the conclusion that any positive integer that is equivalent to 1 mod 6 (i.e. 6n+1) or 5 mod 6 (i.e. 6n-1) COULD POSSIBLY be prime.
This is because not all numbers of the form 6n + 1 or 6n + 5 (i.e. 6n-1) are prime, but definitely every prime that is greater than three will be equivalent to one of those two forms.

Hope I've been reasonably clear in the above steps

Originally posted by tauristercus
reply to post by stander

 

That's strange that you couldn't grasp the meaning of "near infinity," when you use the term "infinity of primes," which is very misleading, coz infinity is not a number, nor is it a synonym to "bunch of" or "plenty of" primes.

Nothing strange about it as in maths, the term "infinity", usually denotes an unbounded limit.
In this thread, we are concerned only with the set of positive integers n, where n grows without bound, i.e. Ip = [1,2,3,4,5, ...., n].

Don't make the common mistake of thinking that the concept of "infinity" can be equivalent to a number, because it can't.

LOL. I don't make mistakes of this kind, you did, and needed to be corrected.

Otherwise, your explanation can be taken for a proof -- no need for rigorous delivery.
Why would I be insistent on the proof?
Well, that's because this way of finding number candidacy for being a prime is more straightforward then the one attached to the prime circle. You just use division with reminder: If the reminder of m/6 is an even number or zero, m is even; if the reminder is odd, m is odd as well. Since primes are a subset of odd numbers, you focus only on reminders 1, 3 and 5. If the reminder is 3, number m must be divisible by 3 and therefore it cannot be a prime. If the reminder is 1 and 5 than there is a possibility that m is a prime. So if m/6 = a reminder 1 or 5, then (m +- 1)/6 = a reminder 0, and therefore m must be divisible by 6.

I don't know why the poster presented this property of primes the way he did, when the division with reminder, the way things were divided in the ancient times, pretty much clues why there is such a property. I guess that's another example of the difference between knowing and understanding.

I found only one reference to the prime circle on the web published in 2003. I think it's not a bad geometric figure to introduce the prime numbers visually. Primes are open subject, but restricted by modern math and issues. I wonder if someone could present the primes in 3D figure or something like that. Maybe I'm going to make a fractal out of them.