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Prime number patterns

jasondavies.com

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Re: Prime number patterns

#81

Post by tryeng »

cturner wrote: post #77
brianpan wrote: post #73

Earlier quoted context omitted.

1. By definition 2. Because the math works out better If it turns out that things that depend on prime numbers work in all cases except for 1, you might as well define 1 not to be a prime number. http://en.wikipedia.org/wiki/Prime_number#Primality_of_one

"By definition" is a poor justification. I almost referenced the exact link you've given here. It gives poor justification for primality of one - resting on authority and lacking justification. The closest it comes (not very) is, "If 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified." So…

I don't believe it is a poor justification. It is a name chosen for this particular set of numbers. In fact, every name we've put on things in mathematics is "by definition". Real numbers, pi, odd numbers, perfect numbers. Someone found that particular definition useful, and others continued to use it.

In the case of prime numbers I believe you cited one of the best reasons of leaving 1 out of the set. If someone needs a name for "the number one and all prime numbers", they can define it.

Re: Prime number patterns

#82

Post by aangjie »

JoeCortopassi wrote: post #48

Stop it, stop it, stop it, just stop. Once an article reaches the front page, it's title is no longer editable. It causes confusion and frustration, and is obviously an issue that a lot of people dislike. I don't care about prime numbers, this article was all about the visualization to me. Every time a title gets changed like this, you are telling your user base that you don't care about what they think. I feel like…

Can't agree more... I care about prime numbers and also about visualizations. But this title is misleading. The article is not about patterns in prime numbers, but a way of visually/geometrically representing prime numbers, which is interesting on it's own. P.S: I don't what was the original title, but the current is definitely misleading.

Re: Prime number patterns

#83

Post by Retric »

jjaredsimpson wrote: post #74
Retric wrote: post #66

Earlier quoted context omitted.

The number between them have a high number of small factors including 2 and 3. 2 * 3 = 6: 5 and 7 are prime. 2 * 3 * 2 = 12: 11 and 13 are prime 2 * 3 * 3 = 18: 17 and 19 are prime 2 * 3 * 5 = 30: 11 and 13 are prime 2 * 3 * 7 = 42: 41 and 43 are prime. However, 2 * 3 * 101 = 606 but 605 is not prime. But, 2 * 3 * 5 * 5 = 150 and 149 and 151 are prime. 2 * 3 * 2 * 3 * 5 = 180 and 179 and 181 are prime.

I don't see where the insight is. twin primes must be of the form (6k-1, 6k+1). So of course there will be a 2,3 at least. Smaller numbers have multiples that are more densely distributed among the integers.

The point is twin primes (6k-1, 6k+1) are more likely for a large k when k is a composite number than a prime AND the more factors of k the higher chance for twin primes.

EX: K = (6 * 2 * 3 * 5 * 7 * 11 * 13 ) gives a twin prime.

Re: Prime number patterns

#84

Post by marshray »

toomuchcoffee wrote: post #58
coderholic wrote: post #53

Earlier quoted context omitted.

The title originally included something like "visualization with D3.js", which has been removed for some reason

...and which was why I clicked on the article in the first place. Because the title was, you know, helpful and descriptive.

I clicked on the title "Prime Number Patterns" and am a little disappointed.

I'm just not seeing any patterns other than the semicircles of increasing integer diameters.

Should I stare at it longer?

Re: Prime number patterns

#85

Post by yaks_hairbrush »

dxbydt wrote: post #42
nessus42 wrote: post #34

Earlier quoted context omitted.

It tells you that prime numbers and periodic curves might be related more than you might have thought at first blush. I.e., if you can mathematically describe and analyze this set of periodic curves, then you have also described and can analyze the prime numbers. As I mentioned, I don't know much about number theory, but I do know that it uses a lot of math that is counterintuitive at first blush. Perhaps this visual…

>if you can mathematically describe and analyze this set of periodic curves, then you have also described and can analyze the prime numbers. Dude, no offense, you are really making stuff up. The periodicity has to to with the divisors of the composites. Every prime p has exactly 2 curves - the wave of period 1, and the wave of period p. There's nothing interesting or useful to take way from that observation. otoh, yo…

> Dude, no offense, you are really making stuff up.

No. Parent has it essentially correct. Many new results in number theory are obtained by studying automorphic forms, which are the stable waveforms, on various spaces.

Things like the Riemann zeta function arise out of spectral transforms of automorphic forms.

Re: Prime number patterns

#86

Post by lloeki »

aaronharnly wrote: post #43

Very, very lovely. You might (or might not) also be familiar with the Ulam Spiral[1] or the arguably more beautiful Sacks Spiral[2], which do reveal certain patterns in the distribution of the primes. I used the software from this site[3] to generate a large Sacks spiral graphic, which I had custom-printed on my shower curtain. To most folks who see it, it's just some pattern of dots, but knowing the order within giv…

Fantastic.

Fig.5 from [2] (Sacks spiral) reminded me of light cones tipping over and pointing at the singularity as one enters a black hole.

[0] http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/bla...

[1] http://plato.stanford.edu/entries/spacetime-singularities/#G...

Re: Prime number patterns

#87

Post by olalonde »

jasondavies wrote: post #39
mikegirouard wrote: post #27

Earlier quoted context omitted.

That is a very nice touch. Good catch. It does seem to break a bit if you resize the browser after the initial render. A refresh will fix it though.

Thanks, it should handle resizing better now.

Slightly off-topic but how did you find out your site was submitted on HN?

Re: Prime number patterns

#88

Post by Confusion »

aaronharnly wrote: post #43

Very, very lovely. You might (or might not) also be familiar with the Ulam Spiral[1] or the arguably more beautiful Sacks Spiral[2], which do reveal certain patterns in the distribution of the primes. I used the software from this site[3] to generate a large Sacks spiral graphic, which I had custom-printed on my shower curtain. To most folks who see it, it's just some pattern of dots, but knowing the order within giv…

  which do reveal certain patterns
Perceived patterns. If an actual pattern was known, you could generate prime numbers from the pattern. No such pattern is known to exist.

Re: Prime number patterns

#89

Post by Confusion »

cturner wrote: post #77
brianpan wrote: post #73

Earlier quoted context omitted.

1. By definition 2. Because the math works out better If it turns out that things that depend on prime numbers work in all cases except for 1, you might as well define 1 not to be a prime number. http://en.wikipedia.org/wiki/Prime_number#Primality_of_one

"By definition" is a poor justification. I almost referenced the exact link you've given here. It gives poor justification for primality of one - resting on authority and lacking justification. The closest it comes (not very) is, "If 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified." So…

"By definition" is the only possible justification, as for any other axiom in mathematics. Your intuition that such a thing is 'poor' and that there is a 'better' (why, more natural?, ordained by a divine source?) way to do it, is wrong.

Re: Prime number patterns

#90

Post by palmtree3000 »

Confusion wrote: post #88
aaronharnly wrote: post #43

Very, very lovely. You might (or might not) also be familiar with the Ulam Spiral[1] or the arguably more beautiful Sacks Spiral[2], which do reveal certain patterns in the distribution of the primes. I used the software from this site[3] to generate a large Sacks spiral graphic, which I had custom-printed on my shower curtain. To most folks who see it, it's just some pattern of dots, but knowing the order within giv…

which do reveal certain patterns Perceived patterns. If an actual pattern was known, you could generate prime numbers from the pattern. No such pattern is known to exist.

It's probabilistic, but there are patterns (lines with significantly greater prime densities than others).
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