5 ms·

Simple wave-optical superpositions as prime number sieves (2019)

34 points by Cieplak 6y ago

Kednicma 6y ago
I am reminded of two curious statements which are not normally linked: (0) There could be a Siegel zero [0], like an "indentation" in the Riemann zeta function. The prime numbers need to all collaborate to "shift" the zeta function a little in order to "buckle" the zeta function at the Siegel zero, so there's only really enough room in the primes to accomodate one Siegel zero at most. (1) Ultrafinitism [1] comes in many flavors, including a flavor where there is a single largest prime number [2]. In this ultrafinite physical world, the largest prime determines how large any object can be before folding back in on itself. Since this would be a true statement about natural numbers, it would also be a sort of conspiracy amongst the primes, with only one instance. Situations like this article remind me of both simultaneously: If there's any sort of conspiracy amongst the primes to forge a Siegel zero, it would have to be the same conspiracy which produces an ultrafinite prime candidate; the primes don't have bandwidth for two conspiracies. [0] https://en.wikipedia.org/wiki/Siegel_zero https://en.wikipedia.org/wiki/Siegel_zero [1] https://en.wikipedia.org/wiki/Ultrafinitism https://en.wikipedia.org/wiki/Ultrafinitism [2] https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/...
- msla 6y ago
  > In this ultrafinite physical world, the largest prime determines how large any object can be before folding back in on itself. This is why I don't like ultrafinitism: It insists on mixing levels, like an Escher print. The world of pure mathematics isn't the physical world, for all the same reasons Santa Claus isn't real, and trying to bind pure math to some (theoretical!) properties of the physical world is just as wrong-headed as insisting that nobody can ever make songs about Santa because we know nobody can deliver toys that quickly.
  - mensetmanusman 6y ago
    But it’s still a useful exercise, because at the interface where math does seem to touch the real world, we can use it to make amazing predictions about the behavior of future events.
deleted 6y ago
[deleted]
kurlberg 6y ago
Low tech approach: Lehmer's bicycle chain sieve: https://en.wikipedia.org/wiki/Lehmer_sieve https://en.wikipedia.org/wiki/Lehmer_sieve
me_me_me 6y ago
I tried to understand this paper but failed miserably, can anyone ELI5 for me?
- lordgrenville 6y ago
  Seconding this. My very basic understanding was that you propagate a wave with a frequency of the first integer, then overlay another wave of the next, and so on. As you go up, eventually the primes appear as the only place where they interfere with each other completely so the wave is at 0.
  - posix_compliant 6y ago
    This is describing the sieve of eratosthenes! Very cool.
- mensetmanusman 6y ago
  Not ELI5, but the position of prime numbers on the number line can be thought of as atoms on a 1 D crystal. The spacing is quasicrystalline. The interesting thing about quasi-crystals, are that they can be modeled as higher dimensional crystals projected on a lower dimension.... (quasicrystals are super stable, which is why we find them in meteorites, we have no known way of manufacturing bulk quasicrystals).

34 points by Cieplak 6y ago

I am reminded of two curious statements which are not normally linked: (0) There could be a Siegel zero [0], like an "indentation" in the Riemann zeta function. The prime numbers need to all collaborate to "shift" the zeta function a little in order to "buckle" the zeta function at the Siegel zero, so there's only really enough room in the primes to accomodate one Siegel zero at most. (1) Ultrafinitism [1] comes in many flavors, including a flavor where there is a single largest prime number [2]. In this ultrafinite physical world, the largest prime determines how large any object can be before folding back in on itself. Since this would be a true statement about natural numbers, it would also be a sort of conspiracy amongst the primes, with only one instance. Situations like this article remind me of both simultaneously: If there's any sort of conspiracy amongst the primes to forge a Siegel zero, it would have to be the same conspiracy which produces an ultrafinite prime candidate; the primes don't have bandwidth for two conspiracies. [0] https://en.wikipedia.org/wiki/Siegel_zero https://en.wikipedia.org/wiki/Siegel_zero [1] https://en.wikipedia.org/wiki/Ultrafinitism https://en.wikipedia.org/wiki/Ultrafinitism [2] https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/...

msla 6y ago
> In this ultrafinite physical world, the largest prime determines how large any object can be before folding back in on itself. This is why I don't like ultrafinitism: It insists on mixing levels, like an Escher print. The world of pure mathematics isn't the physical world, for all the same reasons Santa Claus isn't real, and trying to bind pure math to some (theoretical!) properties of the physical world is just as wrong-headed as insisting that nobody can ever make songs about Santa because we know nobody can deliver toys that quickly.
- mensetmanusman 6y ago
  But it’s still a useful exercise, because at the interface where math does seem to touch the real world, we can use it to make amazing predictions about the behavior of future events.

deleted 6y ago

[deleted]

kurlberg 6y ago

Low tech approach: Lehmer's bicycle chain sieve: https://en.wikipedia.org/wiki/Lehmer_sieve https://en.wikipedia.org/wiki/Lehmer_sieve

me_me_me 6y ago

I tried to understand this paper but failed miserably, can anyone ELI5 for me?

lordgrenville 6y ago
Seconding this. My very basic understanding was that you propagate a wave with a frequency of the first integer, then overlay another wave of the next, and so on. As you go up, eventually the primes appear as the only place where they interfere with each other completely so the wave is at 0.
- posix_compliant 6y ago
  This is describing the sieve of eratosthenes! Very cool.
mensetmanusman 6y ago
Not ELI5, but the position of prime numbers on the number line can be thought of as atoms on a 1 D crystal. The spacing is quasicrystalline. The interesting thing about quasi-crystals, are that they can be modeled as higher dimensional crystals projected on a lower dimension.... (quasicrystals are super stable, which is why we find them in meteorites, we have no known way of manufacturing bulk quasicrystals).