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Message boards : General discussion : Is there a pattern to prime numbers?

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AT Hiker

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Message 65891 - Posted: 25 May 2013 | 18:55:30 UTC

The answer to subject question I assume is no since if there was, finding all prime numbers would be simple.

But, can we be sure there is no pattern to prime numbers?

PS I am not a math person!
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Message 65894 - Posted: 25 May 2013 | 19:52:12 UTC - in response to Message 65891.

I also question that question to myself :)
And if pattern exist it is beyond our knowledge.
And I found "pattern" in two numbers...
look at this

1 pair
15*2^2785940+1
17*2^2721830-1

2785940-2721830= 64110

2 pair
15*2^2988834+1
17*2^2946584-1

2988834-2946584= 42250

3 pair
15*2^3162659+1
?
So if you follow pattern 64110, 42250, then third one will be in range 20000-30000
so next 17*2^N-1 will be in range 3132659 to 3142659
In that range is about 373 candidates so that can be processed in day or two..
good luck! and inform if you find something :)

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Message 65898 - Posted: 25 May 2013 | 21:00:59 UTC - in response to Message 65894.

There are many patterns to PROBABLE prime numbers.

1) All prime numbers greater than 3 are a multiple of 6 +- 1.
Proof : any number can be written as 6n+{-1,0,1 2 3 4), as 6 is even 6n is even and 6n+{0,2,4} is even and not prime, 6 is divisible by 3 so 6n+{0,3} cannot be prime, leaving 6n+-1 as possible prime numbers. Not all numbers 6n+-1 are however prime, e.g. 25 is composite but 6*4+1.

Note 6n+{x,y...} means 6 times something plus one of the numbers in the braces.

2) Euclid proved that if a continuous series of prime numbers are multiplied and then 1 is added the result will be prime, this is the original proof of an infinite numbers of primes. Not all primes fit into this series though.

Many other more maths based theories are out there. It is on these that the LLR method is based for example, the study of these is considered by some to be one of the most interesting subjects in maths.
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AT Hiker

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Message 65909 - Posted: 26 May 2013 | 3:27:31 UTC

How about a pattern involving the spacing of prime numbers?
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[DPC]Camulos

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Message 65914 - Posted: 26 May 2013 | 8:25:26 UTC

@AT Hiker: If you could predict spacing between prime numbers; then primegrid would become obsolete.. and prime numbers with billions of digits could be calculated.

To cheer you up.. there are several visualizations for prime number gaps :)
Check out:
- Ulam Spirals
- Sieves of Chaos
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Message 65967 - Posted: 27 May 2013 | 11:42:13 UTC - in response to Message 65909.

AT Hiker wrote:
How about a pattern involving the spacing of prime numbers?

I just read an article about Zhang's Theorem on Bounded Gaps Between Primes stating he proved there will always be prime gaps < 70 million. Too large to make PG obsolete...
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AT Hiker

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Message 65981 - Posted: 27 May 2013 | 14:39:12 UTC

Thanks to all who replied.

The visualizations on Primal Chaos site are really neat.

I assumed from the start that there was no "known" pattern to the spacing of primes but I also wondered if the spacing pattern might be very, very long and we haven't reached the end yet.

Has someone "proved" that a pattern will never occur?

Is anyone actually analyzing the spacing?

Thanks again.
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Herr Roedy

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Message 66219 - Posted: 31 May 2013 | 9:15:41 UTC

Well, recently a pattern has been proven:
The gap between 2 prime numbers is never greater than 70 million. Truly remarkable, considering how numbers can get infinitly large.

Btw, if you're into real hardcore research, try using google scholar. You can get acces to a lot of scientific papers and if you're a student you can probably acces the full version of all papers for free.
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Message 66220 - Posted: 31 May 2013 | 9:47:34 UTC - in response to Message 66219.

Well, recently a pattern has been proven:
The gap between 2 prime numbers is never greater than 70 million. Truly remarkable, considering how numbers can get infinitly large.

Zhang did not prove that the gap is < 70M for every pair of consecutive primes, but just that there are infinitely many such pairs. It's still possible (and in fact quite likely) that there are gaps > 70M.
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Message 66234 - Posted: 1 Jun 2013 | 1:39:58 UTC - in response to Message 66220.
Last modified: 1 Jun 2013 | 1:40:39 UTC

Not just quite likely but definitely exist and easily constructed.
Consider all numbers 70000002!+k with 2<=k<=70000002. All composite.

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Message 66247 - Posted: 1 Jun 2013 | 14:20:40 UTC - in response to Message 65981.

...
I assumed from the start that there was no "known" pattern to the spacing of primes but I also wondered if the spacing pattern might be very, very long and we haven't reached the end yet...

I've noticed a pattern. Prime numbers never end in 4 , 6, 8 or 0 and hardly ever end in 2 or 5.

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Message 66274 - Posted: 2 Jun 2013 | 7:17:48 UTC - in response to Message 66247.

wrong answer :-)
some numbers always end with those numbers :-)
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Message boards : General discussion : Is there a pattern to prime numbers?

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