269 — The structure factor of primes

Zhang, Martelli, & Torquato (1801.01541)

Read on 16 May 2018
#primes  #number-theory  #Bragg-peaks  #crystals  #crystallography  #fractals  #math 

I usually stay relatively close to my comfort-zone when picking new papers to read. Even if I pick a totally new topic, it often has some bearing on something that I’ve read or have experience with in the past.

So this paper is a bit of an exception — I’m very much outside of my comfort zone when it comes to pretty much anything having to do with prime numbers, outside of a very perfunctory understanding of what they are and why they’re weird.

But prime numbers are very weird — their occurrence along the real number line follows a sequence that, despite our best efforts, we’ve so far completely failed to map in a comprehensive way. (That’s not to say we haven’t tried, or that we know nothing!)

And so clever ways of identifying and better understanding primes is a fascinating area of research. I always like seeing what interesting strategies people develop for primality tests or prime-probability tests: The Prime Spiral is a very cool illustration of how this semi-random looking system has high-level order.

The authors of this paper found another fascinating strategy: Pretend that numbers are infinitesimal dots of refractive material, shine a virtual light on them, and look at the radiation output.

In the real world, crystallographers use Bragg peaks — peaks along the intensity of this refractive output — to understand the microstructure of crystal lattices, even when looking at individual crystals is impossible. The result of this paper suggests that a similar method can be used to identify the 1D structure of prime number distributions across the number line: Fascinatingly, the ratio of any two Bragg-like peaks in this simulation is rational, which suggests that there is some fundamental pattern to the location of primes (though the function to derive this may still be complicated).