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Prime number patterns

jasondavies.com

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Re: Prime number patterns

#91

Post by Lewton »

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…

Peculiar how this comment is at the bottom of this page, yet has enough upvotes to have made it to the best comments page http://news.ycombinator.com/bestcomments

Re: Prime number patterns

#92

Post by monochromatic »

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.

There are TONS of patterns in the primes.

Re: Prime number patterns

#93

Post by arunix »

cturner wrote: post #70

It's interesting that the curve on the left passes through zero rather than one. I've always been inclined to think 1 x = x, and think of one as fundamental. But the pattern shows that rather than 1 x being fundamental, rather there is the issue of x exists or x doesn't exist. x doesn't exist => 0. I wonder if 1*x = x is really a distraction away from a better type of thinking around existence. Also, I've always been…

Visually, non-primes can be represented as a symmetrical shape

e.g. 6 can be shown as

  xxxxxx
but also as

  xxx
  ---
  xxx
or

  x|x
  x|x
  x|x
Whereas a prime like 5 doesn't have this symmetrical layout. Since 1 does have this kind of symmetry it belongs more with the non-primes.

Re: Prime number patterns

#94

Post by Confusion »

monochromatic wrote: post #92
Confusion wrote: post #88

Earlier quoted context omitted.

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.

There are TONS of patterns in the primes.

Name me a pattern that, from the primes below 100, can be used to extrapolate, with certainty, the first prime above 100.

Re: Prime number patterns

#95

Post by nessus42 »

yaks_hairbrush wrote: post #85
dxbydt wrote: post #42

Earlier quoted context omitted.

>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.

Thank you. I don't know what's up with the increasing trend around here for people to imply you are an idiot over some nitpick that seems to reveal only that the nitpicker spent no effort to try understanding what you had to say, and would rather berate you for a detail rather than engaging in the gist.

One of the things that can seem almost mystical at times about math to someone who has not studied math heavily, is even just simple things, like how pi and e seem to get into everything, even where you might not naively expect it.

The visualization in the OP shows how to use sine waves to build a sieve of Sieve of Eratosthenes. Now that I've seen the visualization, this revelation seems so utterly obvious that it goes without saying. But somehow, I never drew this connection until seeing the visualization.

And once I see how "obvious" this is, it's suddenly obvious how e and pi might get into everything, because everything that repeats with a specific frequency can be modeled as a wheel rolling along and leaving a mark on every revolution. And what is multiplication, but repeated addition? I.e., a wheel of a certain size rolling down the number line, leaving its mark once per turn. Above a certain age, we tend to stop thinking about multiplication as repeated addition, and so we don't think about how all multiplication is implicitly bringing pi into everything we are doing.

Maybe everything I said above is wrong in some way, since, as I have mentioned, I haven't studied any math past calculus and college algebra, and even that was so long ago, most of it I don't remember. Or maybe what I've said is so obvious to someone who has studied math seriously that they just want to shout, "Duh!" But there must be some way to interpret what I just wrote that doesn't deserve being summarily shot down.

Re: Prime number patterns

#96

Post by monochromatic »

Confusion wrote: post #94
monochromatic wrote: post #92

Earlier quoted context omitted.

There are TONS of patterns in the primes.

Name me a pattern that, from the primes below 100, can be used to extrapolate, with certainty, the first prime above 100.

That's an awfully restrictive definition of "pattern." For what it's worth, I'd say that more mathematics has been devoted to the study of patterns in the primes than to any other single topic.

Re: Prime number patterns

#97

Post by yaks_hairbrush »

nessus42 wrote: post #95
yaks_hairbrush wrote: post #85

Earlier quoted context omitted.

> 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.

Thank you. I don't know what's up with the increasing trend around here for people to imply you are an idiot over some nitpick that seems to reveal only that the nitpicker spent no effort to try understanding what you had to say, and would rather berate you for a detail rather than engaging in the gist. One of the things that can seem almost mystical at times about math to someone who has not studied math heavily, is…

You're welcome!

> One of the things that can seem almost mystical at times about math to someone who has not studied math heavily, is even just simple things, like how pi and e seem to get into everything, even where you might not naively expect it.

Yep. Sure blew my mind when I saw a proof of quadratic reciprocity (a very neat result about square numbers in modular arithmetic) which used complex analysis (how on earth can complex numbers prove stuff about modular arithmetic!?)

> Maybe everything I said above is wrong in some way...

Not really. Your intuition upon seeing this visualization was pretty much right on: studying periodic functions is a way of understanding numbers.

> Or maybe what I've said is so obvious to someone who has studied math seriously that they just want to shout, "Duh!"

Not so much. It took some mighty smart folks to develop some ideas which are perhaps suggested, in hindsight, by this picture. The big one is Fourier series and transforms, which allow you to decompose periodic functions into their constituent sine waves. You can use Fourier analysis to get information about number theory, which was essentially your suggestion. However, that's not at all obvious without seeing this picture. Certainly, my first exposures to Fourier analysis were in the context of signal processing and solving PDEs. I had absolutely no inkling that it may be useful for number theory until actually seeing it. Even if I had seen this picture 7 years ago (when I knew signal processing and PDEs, but not number-theoretic applications), I probably would not have made the connection that you made.

So, I think your intuition was a rather non-obvious idea, and so your comment did not deserve the quick shoot-down. (And even if it were obvious to folks who have studied math, it would still be non-obvious to someone, probably).

Re: Prime number patterns

#98

Post by toomuchcoffee »

marshray wrote: post #84
toomuchcoffee wrote: post #58

Earlier quoted context omitted.

...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?

Umm... yes, you should actually.

Those patterns of semicircles aren't random, of course. They correspond directly to the degree of compositeness of the chosen modulus. Compare for n = 60,61,62, for example.

The higher the totient value for n, the more circles you see, basically.

Re: Prime number patterns

#99

Post by [deleted] »

nessus42 wrote: post #95
yaks_hairbrush wrote: post #85

Earlier quoted context omitted.

> 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.

Thank you. I don't know what's up with the increasing trend around here for people to imply you are an idiot over some nitpick that seems to reveal only that the nitpicker spent no effort to try understanding what you had to say, and would rather berate you for a detail rather than engaging in the gist. One of the things that can seem almost mystical at times about math to someone who has not studied math heavily, is…

[deleted]

Re: Prime number patterns

#100

Post by nessus42 »

[deleted] wrote: post #99
nessus42 wrote: post #95

Earlier quoted context omitted.

Thank you. I don't know what's up with the increasing trend around here for people to imply you are an idiot over some nitpick that seems to reveal only that the nitpicker spent no effort to try understanding what you had to say, and would rather berate you for a detail rather than engaging in the gist. One of the things that can seem almost mystical at times about math to someone who has not studied math heavily, is…

[deleted]

Are you claiming that I've gleaned no insight at all as to why Fourier transforms are used in number theory? Or are you claiming that I just didn't acquire this insight from the visualization?

If you claim the latter, then just where did I get it from?

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