Originally posted by mathfreak
Finding all the squares of primes (except 4 and 9) on line 1 is really cool, something I didn't know, and I salute you for it, but there's a reason for this, too. Modular arithmetic says we only need to check the squares of the first circle. If they all lie on line 1, then all possible prime squares will lie on line 1.

It's pretty easy to see why. Consider "p^2 - 1", where "p" is prime. The value will always be an even even multiple of 24. This is because "p^2 - 1" can be factored as "(p-1) * (p+1)".

Consider any sequence of three contiguous integers: "(x-1), x, (x+1)".

Two of them must be divisible by two (even numbers). One of them must be divisible by three, and one of them must be divisible by four. In the case of a prime number "p", which of course is not divisible by anything, then we can say: one of (p-1) or (p+1) is divisible by four, the other by two. Additionally, one of them is divisible by three.

Thus, "p^2 - 1" always has the factors 2, 4, and 3, and is divisible by 2*4*3 = 24. That's why it's always on the same radial line.

Originally posted by stander
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]

@stander, the problem is that this method doesn't eliminate all of the composite numbers, only some of them. And if I knew how to do this, I wouldn't post it here I'd publish it or use it to break public key crypto systems and become rich.

However, with that said, I have done some interesting work with prime number patterns. First hint: note that the digits of prime numbers are not uniformly distributed, that is, they do not occur with equal probability.

OP; Very interesting stuff, kept me out of the Pub this evening.

This thread caught my attention, so I delved in without much in the way of research I'm afraid.

Using my MS based PC and Excel I wrote some simple 'macros' and graphed the results:

For 1120001- 1130000 I find 718 primes
For 2120001- 2130000 I find 697 primes

For 12120001-12130000 I find 604 primes
For 22120001-22130000 I find 596 primes
For 32120001-32130000 I find 565 primes
For 42120001-42130000 I find 560 primes

For 142120001-142130000 I find 539 primes
For 242120001-242130000 I find 523 primes

For 1242120001-1242130000 I find 488 primes
...if I go above the 'Long' positive in Excel I get an error, so my probing ends there.

Looking at these numbers would imply that rather than 'infinate', that there is an ultimate prime number. The number of primes, -for my 10,000 selections-, decreases as the value increases. That would imply to me that there was an eventual destination?

Does the number of primes decrease as the value grows?

Is there a log or log-log graph available for view that records this as it runs beyond my capability?

No progress then.

mmm... I'm sure I saw something that rang a bell in another post.

It was someting about 'A heartbroken father shows us how to protest' I think it was father of Lance Corporal Shaun Brierley, murdered for MONEY.

Whether your idea is new or not, or can be explained by modular arithmetic or not, I find it a fascinating topic and more fascinating you found this out by yourself! So S&F from me for this great thread and your good explanations!

I'm making an Excel sheet now, putting it in a 24-column table instead of a circle, which basically works the same. Let's see if we can come up with a method to calculate the difference in row or circle position. Useful or not, it's a very interesting challenge that might shake the modern cryptographic cradle on it's foundation.

Keep up the great work!

[edit on 24-5-2010 by Professor Chaos]

DIAGRAM1:





Not sure how to quote properly


I have one question regarding the above image and the others like it in your thread further down.

Why have you disregarded 2 and 3 as prime numbers?

At the top of the page you had them included in the list but not number 1, however when you arrived at the diagram above you included 1 but not 2 and 3.



[edit on 24-5-2010 by Professor Chaos]

I am sorry to disappoint you but you idea is not really new. It's basically the same principle Gauß used in the 19th century for his clock-calculator. So while it is still useful it is just something very basic.
The current way mathematicians think we will understand prime numbers some day is Riemann's Conjure. It's implications are really mind boggling. Some arrive at numbers so big that they would have more digits than there are atoms in the known universe.

You can read all about it in the Book "Music of the Primes" from Marcus du Sautoy.
(The title is a bit misleading it's really mainly about Riehmann's work.)


I personally belive that the secret of primes is not just about the prime numbers themselves but also about all other natural numbers. Euler has done some pretty amazing stuff in inventing a formular which can derive somthing like a 'prime equivalent' for every number. While this formula can be used to detemine the sonic quality of rational musical intervals its result yield the number itself if it is prime and a lower number if it is not. Here are the results for numbers 1 to 24
1 1
2 2
3 3
4 3
5 5
6 4
7 7
8 4
9 5
10 6
11 11
12 5
13 13
14 8
15 7
16 5
17 17
18 6
19 19
20 7
21 9
22 12
23 23
24 6

You can lookup the "gradus suavitatis" funtion if you are interested. The really interesting thing would be calculating it the other way around, alias calculating the all the numbers which would result in a particular gradus value. The problem here is that there is no programming language I know of which can handle multidimensional arrays with variable magnitude. Without it the memory requirement would be much too high for practical applications.