Surprising hidden order unites prime numbers and crystal-like materials

Link to paper 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

And here’s some marhoverflow discussions on Dyson’s ideas: https://mathoverflow.net/questions/133581/quasicrystals-and-...

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?

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.

Pardon my extreme ignorance of the subject, but could you elaborate on why your statement is mutually exclusive of the OP? Not at all combative, just a sincerely interested layman :)

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.