Prime After Prime | Hacker News Reader

Prime After Prime

bit-player.org

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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 in 2d and 3d, and altering the way they're displayed (spirals, ovals, wrapping them as square spirals etc) and seeing if anything comes out. I mean, on the off chance that it might be easier to detect a pattern at a human level than apply strict mathematical techniques. Is this still an area under active research, or was it rather quickly tried and then given up with "nope, this isn't getting us anywhere"?

As you can probably tell, I'm not a mathematician. I just have a sort of superstitious belief that there's something "natural" about primes, and that there'll be a pattern there somewhere. I've read about people with mental health issues who somehow are able to tell if a number is prime and assuming that this is accurate then again, unless they are somehow able to run a learned algorithm in their heads stupidly fast they're therefore tapping into something natural. Or perhaps these people weren't assessed by anyone who actually knew what a prime number was.

That could make a pretty neat crowdsourcing project. Come up with a parameterized set of prime number visualizations. Have a website that draws ("draw" in both the sampling and graphical sense) pairs of visualizations, asking the visitor to select which is most "interesting".

When I was a kid, my dad had a postcard lying around with a reply from Isaac Asimov about a pattern my dad said he found in primes by writing them in a table or something. The answer was basically, "when you look at the page your visual cortex is doing a ton of math you're not factoring in..."

https://en.wikipedia.org/wiki/Ulam_spiral

there are a number of ways to answer your question, so consider the following only one:

i do research into the underlying pattern of primes, and do so from the intended inference that there does exist an algorithm that returns primes based on configurable inputs

> starts of saying there's no pattern, then shows a pattern

there are many patterns in the prime number sequence

here is a seemingly ludicrous pattern in the sequence of primes 468,395,662,504,823 + 205,619 · 223,092,870 · n, for n = 0 to 23 (o)

but i think when people say things like: If there’s a pattern, it’s inscrutable; from the article they are referring to an ability to answer questions for all primes such as: what is the three thousandth and one prime after 17, or what prime follows and precedes 113; those kinds of questions are currently unanswerable for all numbers efficiently

> How hard have people looked for pattern visually?

the concept of visual is very subjective

numbers are abstract concepts and how you choose to visualise them is simply a matter of choosing how the abstraction you use will appear, i can make any prime number appear anywhere in a visualisation depending on how i configure the visuals

> the way they're displayed (spirals, ovals, wrapping them as square spirals etc)

dreaming up display methods, and developing them, is math

there is a famous question in mathematics called the riemann hypothesis(i)(ii) where one could argue that complex numbers become a clever abstraction for visualising of primes

> I've read about people with mental health issues who somehow are able to tell if a number is prime and assuming that this is accurate then again, unless they are somehow able to run a learned algorithm in their heads stupidly fast they're therefore tapping into something natural.

confirming if a number is prime or composite, what is called a primality test(iii), is a 'solved' problem with an efficient algorithm

(o) https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem

(i) https://www.youtube.com/watch?v=d6c6uIyieoo

(ii) https://www.youtube.com/watch?v=VTveQ1ndH1c

(iii) https://en.wikipedia.org/wiki/Primality_test

There are certainly patterns. Try this:

Draw lines from the origin with slopes 1, 1/2, 1/3, 1/4, ...

A number, y=n, is prime if no line intersects y=n at an integer other than n.

You can also start out with a line of points, then repeatedly place a new line of points, translated up one unit and right one unit from the start of the last line and increase the distance between each point by one from the last line. [There are many other simple constructions and many repeating sub-units with definite patterns.]

           *         *
          *        *
         *       *   
        *      *      *
       *     *     *
      *    *    *    *
     *   *   *   *   *
    *  *  *  *  *  *  *
   * * * * * * * * * * 
  *********************

The primes are the columns with only two filled points.

http://divisorplot.com records more patterns.

I currently have a demo [0] up that lets you explore these drawings in projective spaces. [Note that the default scene is showing only a scaled portion of the underlying data.]

[0] http://blakelapierre.github.io/jsxbox/

I think the author was trying to say that there seems to be no simple pattern. The original sequences are meant to demonstrate how hard it is to see a pattern. However, a relatively recent empirical discovery found very simple statistical biases between consecutive primes. Mathematicians were shocked by these results because they were so simple. This article demonstrates and further explores this surprising phenomenon.

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.

>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.

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

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.

> 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.

Let p-prime. Consider 3 "probabilities": a) p+2 is prime; b) p+4 is prime; c) p+6 is prime. Obviously p(c)=2p(a)=2p(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) is 0.02. And P(k + 30 is a multiple of 10) is also still 0.1.

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.

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