
Surprising hidden order unites prime numbers and crystal-like materials - foxes
https://phys.org/news/2018-09-hidden-prime-crystal-like-materials.html
======
AGoodName
A brief explanation of why primes peak at repeated multiples from a layman
who's wondered why before.

Obvious first example all primes above 2 are of the form 2x+1. An obvious
repeating pattern of primes.

You can take this a small step further. All primes above 6 are of the form
6x+1 or 6x+5. Anything else is a multiple of 2 or 3. Above 6 only 1/3 of
numbers are worthy of being considered prime. This is a slightly less obvious
example.

A small step further - all primes above 30 are of the form 30x+1, 30x+7,
30x+11, 30x+13, 30x+17, 30x+19, 30x+23 or 30x+29. Anything else is a multiple
of 2,3 or 5. So above 30 only 8/30 numbers are worthy of being considered
prime. See how we've created a new pattern for the multiple of 2x3x5 to rule
out a swath of prime candidates..

I could repeat this each prime found. eg. I could take the common multiple of
2,3,5,7 (210) and create a similar pattern for all numbers above 210 that
rules out the repeated multiples of 2,3,5 and 7. (leaving us just 58/210
numbers worthy of being considered prime).

This is why you see peaks of primes at various repeating multiples. For every
new prime found you can take the multiple of it and all previous primes. From
that you can rule out primality for various offsets to any multiples of that
number. So primes above certain numbers can only possibly exist in certain
forms. Which is why you see primes at repeated patterns from each other - the
primes can only exist in those forms.

~~~
chongli
This pattern forms the basis for wheel factorization [1], a faster way to
factor a number than naïve trial division.

[1]
[https://en.wikipedia.org/wiki/Wheel_factorization](https://en.wikipedia.org/wiki/Wheel_factorization)

~~~
akira2501
Also, the
[https://en.wikipedia.org/wiki/Sieve_of_Atkin](https://en.wikipedia.org/wiki/Sieve_of_Atkin)

~~~
AGoodName
Author of this thread here. That article states that the above sieve was
created in 2003.

I actually wrote and described it back in 2002 -
[https://forums.overclockers.com.au/threads/show-
off.71630/#p...](https://forums.overclockers.com.au/threads/show-
off.71630/#post-813807)

I didn't think much of it back when i did it and i'm not a mathematician, just
a hobbyist. I guess i should publish my prime hobbies more...

~~~
gtt
Have you though about mailing it to authors of the paper? The should somehow
cite you, I guess.

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credit_guy
Link to paper
[https://arxiv.org/abs/1802.10498](https://arxiv.org/abs/1802.10498)

Also worth noting that Freeman Dyson pointed out the link between prime
numbers, or the Riemann zeta function, and quasicrystals, see p. 2-3 of
[http://www.ams.org/notices/200902/rtx090200212p.pdf](http://www.ams.org/notices/200902/rtx090200212p.pdf)

And here’s some marhoverflow discussions on Dyson’s ideas:
[https://mathoverflow.net/questions/133581/quasicrystals-
and-...](https://mathoverflow.net/questions/133581/quasicrystals-and-the-
riemann-hypothesis)

------
karmakaze
I don't find it at all surprising that we find a natural occurrence of prime
number patterns in nature. It makes perfect sense to find them in crystals if
you think about it for a while.

First remember that 'primeness' is not really a property of a number, it's the
absence of the property of being composite. "Can only be divided..." meaning
"Can't be divided..." meaning "Not composite."

Second crystals are formed by repeating patterns. What happens if you compose
a pattern from many repeating patterns and overlay them? There would be
'features' where the patterns don't overlap.

Do papers try to make things seem difficult, exciting and mysterious on
purpose?

~~~
thaumasiotes
> First remember that 'primeness' is not really a property of a number, it's
> the absence of the property of being composite.

What do you mean by this? What do you believe a "property" is?

> Do papers try to make things seem difficult, exciting and mysterious on
> purpose?

Yes, being surprising/exciting is a criterion for being published.

~~~
karmakaze
Certainly they could and are both (prime, composite) described as properties.
The difference is that one can be described positively as 'having' and the
other as 'not having' or 'having only'. If you've read Godel Escher Bach, it's
very much in how it describes axiomatic space with 'foreground' vs
'background' when looking at the boundary of what's inside and outside a set
of a given property. Compositeness is a construction. Primeness is what's
outside.

A sort of example from the book that plays in the "backgroumd':
[https://en.wikipedia.org/wiki/Berry_paradox](https://en.wikipedia.org/wiki/Berry_paradox)

------
eganist
Reminds me, whatever happened to the research into parallax compression?

[http://www.novaspivack.com/science/we-have-discovered-a-
new-...](http://www.novaspivack.com/science/we-have-discovered-a-new-pattern-
in-the-prime-numbers-parallax-compression)

------
oelmekki
I read once that prime numbers were a key element element on cryptography
(because it's easy to multiply two prime numbers, but difficult to say if a
number is a multiple of two prime numbers, if I remember correctly). Will this
discovery have negative impact on it?

~~~
nl
No.

It's the non-factorability of primes that is important.

~~~
gonzo
> It’s the non-factorability of primes that is important.

Primes _by definition_ have two, and only two factors.

The difficulty of factoring a number which is the product of two large primes
is the important bit.

~~~
toxik
I think the usual abstract algebra definition is one factor: itself. 1 is not
considered a factor so that prime factorizations are unique, otherwise you
could tack on an infinity of ones.

Also Möbius becomes strange with infty factors.

~~~
ginnungagap
The abstract algebra definition is that p is a prime if whenever p divides a
product ab then p divides at least one of a and b.

Having no proper factors is the definition of an irreducible element.

The two definitions agree for the integers and nice algebraic structure (UFDs)
but there are algebraic structures in which they are not the same which
explains why there are two differenr definitions and names in abstract algebra

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waynecochran
No mention of whether this knowledge can be exploited to factor integers? You
know where I’m going with this...

