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New Pattern Found in Prime Numbers

05/11/2009 7:07 AM

(PhysOrg.com) -- Prime numbers have intrigued curious thinkers for centuries. On one hand, prime numbers seem to be randomly distributed among the natural numbers with no other law than that of chance. But on the other hand, the global distribution of primes reveals a remarkably smooth regularity. This combination of randomness and regularity has motivated researchers to search for patterns in the distribution of primes that may eventually shed light on their ultimate nature.

http://www.physorg.com/news160994102.html

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#1

Re: New Pattern Found in Prime Numbers

05/11/2009 10:50 AM

...did you expect a reply on your post...?

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#8
In reply to #1

Re: New Pattern Found in Prime Numbers

08/10/2009 9:10 PM

Meh

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#2

Re: New Pattern Found in Prime Numbers

05/11/2009 12:48 PM

Unfortunately because of poor credit I took out a sub-prime number.

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#3

Re: New Pattern Found in Prime Numbers

05/11/2009 12:53 PM

cool post, and timely - I just finished reading In Code, about a Scottish teenager trying to improve on the RSS algorithm

Fun read and I highly recommend to all.

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#4

Re: New Pattern Found in Prime Numbers

05/12/2009 6:40 PM

Every time I read this type of article I keep wondering why the Feigenbaum Number (related to chaos theory) is so closely approximated by the value of 10.0/(Pi-1)

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#6
In reply to #4

Re: New Pattern Found in Prime Numbers

05/13/2009 1:40 PM

Ironically given that the origin is chaos theory, the answer must be pure chance.

Given any number, you can often find a simple formula using integers and well-known irrational numbers (including e, pi, square-roots of moderately-sized integers) that is within a few hundred ppm. This one just happens to be within 50 ppm.

We might cry "major coincidence" if you find a simple formula with a similarly close correspondence for the second Fiegenbaum constant (α ≈ 2.5029078751)

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#7
In reply to #6

Re: New Pattern Found in Prime Numbers

05/13/2009 1:50 PM

Get yourself a moniker and hang around

It would be nice to have another mathematician around - "Physicist?" keeps kicking my tail on puzzles

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#5

Re: New Pattern Found in Prime Numbers

05/13/2009 1:20 PM

The "new pattern" that is in the reference is not a property of the primes as such* - it is a property of the way the the authors selected the primes. The authors point to this as a possible method for detecting fraud, which may possibly be helpful in some circumstances.
*A very similar pattern would be found for integers selected using the same algorithm.

All that has happened is that the authors (or more likely the reporters) have misunderstood the background.

First, to repeat your points with a slightly different emphasis: one of the simplest things that can be said about primes is that the proportion of numbers that are prime reduces as the numbers get larger. The next thing is that the distribution appears unpredictable (other than by exhaustive search).

So far as I know, the so-called regularity is pure red herring - in that no regular patterns have been found that are additional to what you would expect from any other random sequence with a well-defined variable density over different regions.

Let us look at what we might expect for numbers over the decade 10000 to 99999. The density only reduces by about 13% between those starting with '1' and those starting with '9'. That is not very significant - and larger primes are even more uniformly distributed over a similar range. So we can say that the distribution of leading digits is essentially the same as it would be for the integers.

Suppose now that you randomly select n in a way that causes the first digit of 'n' to have a Bentfield distribution. In contrast, the first digit of the nth primes will be approximately uniformly distributed.
To repeat - the property that the authors have found is a result of the selection method, and not a fundamental property of the distribution of the primes themselves.

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