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Prime After Prime

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Re: Prime After Prime

#11

Post by lohankin »

geofft wrote: post #8
lohankin wrote: post #6

Let p-prime. Consider 3 "probabilities": a) p+2 is prime; b) p+4 is prime; c) p+6 is prime. Obviously p(c)=2 p(a)=2 p(b). Probability of p+30 being prime is even higher.

I assume you mean P(c) = P(a) P(b) + P(b) P(a) (i.e., you can get to p+6 as p+2+4 or as p+4+2) = 2 P(a) P(b)? But even so, that's not true, because you're messing with conditional probabilities in a weird way. Let k be any integer. Consider the probabilities a) k+2 is a multiple of 10, b) k+4 is a multiple of 10, and c) k+6 is a multiple of 10. P(a), P(b), and P(c) are all 0.1, but your approach would claim that P(c)…

No, I meant something different. Let's consider primes, say, between 1 and 1000000. a) Compute the number of cases when p is prime AND p+2 is prime; b) number of cases when p is prime AND p+4 is prime; c) number of cases when p is prime AND p+6 is prime. My claim is that a is approx. equal to b, and c is approx. equal to 2a (which is the same as 2b). You can easily test it (which I did), and it's easy to see why.

Re: Prime After Prime

#12

Post by rmetzler »

> A puzzling article; starts of saying there's no pattern, then shows a pattern.

True random samples don't show only equal distribution when you choose single numbers, they also show equal distribution when you get the single numbers in sequences of two or three.

This is a simple test to find out if they are truly random or not. You see the result in these diagrams as the author used x and y of a pair from the modulo sequence of primes.

Re: Prime After Prime

#13

Post by goldenkey »

I'm just going to link to this post: http://math.stackexchange.com/questions/311610/modified-eule...

Primes have a pattern, just obvious by their definition. They are self-similar. That doesn't mean there will be some magic non-linear formula for generating primes or detecting primality. But it does mean there are properties that might not be so obvious.

Re: Prime After Prime

#14

Post by heisenbit »

dave2000 wrote: post #2

A puzzling article; starts of saying there's no pattern, then shows a pattern. In the hope that people who know more about primes that me (this will not be unlikely) are reading this: How hard have people looked for pattern visually? I saw a few pieces about it online but I'd have thought that it would be rather easy to take the first, say, 10 million primes and subject them to all sort of techniques displaying them…

The article starts out a bit slow in a convoluted way which may be off-putting.

There has been recently a major surprising discovery related to consecutive primes (prepub 11 Mar 2016 http://arxiv.org/abs/1603.03720). The value of the article here imho. compared with the referenced original paper (lots of math formulas) is that visualization may help exploring the topic. The author is showing some results of his exploration and is inviting us to join by sharing his code.

Re: Prime After Prime

#15

Post by Someone »

goldenkey wrote: post #13

I'm just going to link to this post: http://math.stackexchange.com/questions/311610/modified-eule... Primes have a pattern, just obvious by their definition. They are self-similar. That doesn't mean there will be some magic non-linear formula for generating primes or detecting primality. But it does mean there are properties that might not be so obvious.

"That doesn't mean there will be some magic non-linear formula for generating primes"

That is correct, but there are magic non-linear formulas for generating only negative numbers and primes. See https://en.m.wikipedia.org/wiki/Formula_for_primes#Formula_b... or http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html

There also is a very simple function that only produces primes, but not all of them: https://en.m.wikipedia.org/wiki/Formula_for_primes#Mills.27_..., https://en.m.wikipedia.org/wiki/Mills%27_constant

Re: Prime After Prime

#16

Post by Grue3 »

dave2000 wrote: post #2

A puzzling article; starts of saying there's no pattern, then shows a pattern. In the hope that people who know more about primes that me (this will not be unlikely) are reading this: How hard have people looked for pattern visually? I saw a few pieces about it online but I'd have thought that it would be rather easy to take the first, say, 10 million primes and subject them to all sort of techniques displaying them…

>A puzzling article; starts of saying there's no pattern, then shows a pattern.

You must've not read until the end because it explains why the pattern appears and that it doesn't actually hold for larger primes.

Re: Prime After Prime

#17

Post by goldenkey »

Someone wrote: post #15
goldenkey wrote: post #13

I'm just going to link to this post: http://math.stackexchange.com/questions/311610/modified-eule... Primes have a pattern, just obvious by their definition. They are self-similar. That doesn't mean there will be some magic non-linear formula for generating primes or detecting primality. But it does mean there are properties that might not be so obvious.

"That doesn't mean there will be some magic non-linear formula for generating primes" That is correct, but there are magic non-linear formulas for generating only negative numbers and primes. See https://en.m.wikipedia.org/wiki/Formula_for_primes#Formula_b... or http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html There also is a very simple function that only produces primes, but not all of them: https://en.…

Thanks for the links. Very interesting stuff. I am definitely going to keep these bookmarked. Mill's constant can be a bit like cheating if it is irrational, because that would mean it is equivalent to packing all the primes into a decimal expansion. But that's just what I read off the surface, I have to see why it's in the form that it is. Just saying, from an information theoretic approach.
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