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Prime Numbers Not So Random?
Jeff Moriarty writes "Some physicists believe they might have caught a whiff of a pattern in the sequence of prime numbers. This would have a huge impact across mathematics, and to people who just really like primes... or like being Prime."
I wonder if this theory could be used to produce code that could be useful for encryption based on prime numbers, such as RSA's work. Would it make it easier to produce reliable prime numbers much larger than 1024 or even 2048 bit? Further, I wonder if this could be used to drastically reduce the time required to brute force an RSA encrypted message. Could the encryption of files that were encrypted with 128 bit technology be rendered all but useless?
You've "easily" proven things by defining them as something. An irrational number is a number with no known, infinite, repeatable sequence? You've *defined* it that way, that doesn't mean you've ever *proven* a number irrational.
Proof that the square root of 2 is irrational: http://everything2.com/index.pl?node_id=928307
Proof that e is irrational: http://everything2.com/index.pl?node_id=930313
Better examples are obviously out there, but I just searched for 'irrational' on E2... You're very ignorant.
I believe posters are recognized by their sig. So I made one.
They don't claim that they have a rule that can create prime numbers, they just claim that prime numbers might not be completely random.
Just like if you have a large prime p, p+210 is 4.375 times more likely to be a prime than a random integer around p. Not a rule, but a hint that primes aren't so random.
Sorry dude, you're wrong.
In the scientific community, proof is established by repeated experimental repetition, in Mathematics, testing this theory lots of times with lots of different numbers (see computers).
Apparently you've just started your degree (or maybe not even yet) so you haven't heard of proof by induction. In proof by induction, you prove that if a statement is true for q=n, it is also true for q=n+1. That's step 1. Then you go and prove that it's true for q=1. Once you've done those two steps the statement is proven for all integer values of q.
Daniel
Carpe Diem
Want the details? Ignore the watered-down article and skip right to the research paper.. all greek to me, but has some interesting plots:
Information Entropy and Correlations in Prime Numbers -- Abstract
Information Entropy and Correlations in Prime Numbers [PDF]
Information Entropy and Correlations in Prime Numbers [Postscript]
-molo
Using your sig line to advertise for friends is lame.
Don't forget the Prime Spiral.
This construction was first made by Polish-American mathematician Stanislaw Ulam (1909-1986) in 1963 while doodling during a boring talk at a scientific meeting. While drawing a grid of lines, he decided to number the intersections according to a spiral pattern, and then began circling the numbers in the spiral that were primes. Surprisingly, the circled primes appeared to fall along a number of diagonal straight lines or, in Ulam's slightly more formal prose, it "appears to exhibit a strongly nonrandom appearance"
More info.
As a number theory graduate student, this looks suspicious. This isn't as bad as last summer, when some string theorists claimed a junk
proof of the Riemann Hypothesis, but it's close.
Prime numbers are very hard to tackle. Part of the difficulty in this style of problem, as another post points out, is that they are defined multiplicatively, and yet we here care about additive properties (differences in this case).
I have a few concerns with this paper:
1. They look at a really small number of primes (only 10^7 of them). Many false conjectures have been made that way. The most famous case is with the prime number theorem: it's known that up to x there are about x/log(x) primes, and as x grows this estimate becomes more and more accurate. If you do some tests you'll quickly see that there are more than x/log(x) primes up to x for all x you can test for. This was conjectured to be true for all x, until someone proved that actually the difference (# primes up to x) - x/log(x) changes sign infinitely often. The first change is known to happen before x=10^370 -- but try testing that.
2. They use the ansatz Alog(log(x))+B to fit some function of x (the entropy). But for x in the range of concern (at most 10^8), log(log(x)) is essentially constant. Try graphing that function and you'll see for yourself. For all practical purposes (i.e. unless you can run your computer up to numbers like 10^100), doing curve fitting with this function is very suspicious.
My take,
Lior
- I can't believe no one even tryed this before they actually published this article.
Well, they did. Thats what all the above gags are about, that these physicists are unaware of prior, basic "Fun with Primes!" work. Nature too, evidently.
The pattern they've found is a logarithmic distribution, it seems, according to their abstract. (I need to make time to read the full paper.) This is not unexpected, the Newcombe distribution known as Benford's Law (1) is a well-known logarithmic distribution applicable to most naturally occurring numbers. Benford's law is
which applies of course only to the non-zero data in the dataset; it generalizes toWhile their reported entropy power law is logarithmic, Benford's law doesn't appear to fit the prime interval increments. This leaves open the power law the physicists have found is different, suggesting the possibility that there is something interesting and new in their finding. We can hope this gives the number theorists some fresh insight. Since it was posted to the Physics :: Condensed Matter directory of arXiv, it may be a while before the real number theorists even notice it; sci.math.research quite reasonably gets postings of the new listings of only the Math subdirs weekly. We can hope Baez cross-posts it.