
Prime After Prime - bit-player
http://bit-player.org/2016/prime-after-prime
======
dave2000
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.

~~~
markisus
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.

------
goldenkey
I'm just going to link to this post:
[http://math.stackexchange.com/questions/311610/modified-
eule...](http://math.stackexchange.com/questions/311610/modified-eulers-
totient-function-for-counting-constellations-in-reduced-
residue/1055504#1055504)

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.

~~~
Someone
_" 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...](https://en.m.wikipedia.org/wiki/Formula_for_primes#Formula_based_on_a_system_of_Diophantine_equations)
or [http://mathworld.wolfram.com/Prime-
GeneratingPolynomial.html](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/Formula_for_primes#Mills.27_formula),
[https://en.m.wikipedia.org/wiki/Mills%27_constant](https://en.m.wikipedia.org/wiki/Mills%27_constant)

~~~
goldenkey
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.

------
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.

------
lohankin
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.

~~~
geofft
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.

~~~
lohankin
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 2 _a (which is the same as 2_ b). You can easily test it
(which I did), and it's easy to see why.

