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There might be none. But I was thinking of links between number theory and physics, and this would seem like an example that would definitely solidify that link.

Are there any known natural systems, or physical systems in general (possibly designed), that test the primality of a number directly using the laws of physics? (Barring obvious examples such as machines intricately designed with interlocking parts/electronics to do such a job)

If there are none, I would not be surprised. But I was just curious.

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  • $\begingroup$ Related: physics.stackexchange.com/q/414/2451 $\endgroup$ – Qmechanic Dec 2 '12 at 15:49
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    $\begingroup$ Evolution came up with 13- and 17-year cicadas. Presumably the ones with non-prime year cycles were out-competed. $\endgroup$ – Peter Shor Dec 3 '12 at 1:34
  • $\begingroup$ I've asked with my account of MathOverflow in the past a post that maybe fits with your question. Please feel free (you or the professors and users of this Physics Stack Exchange) to provide your feedback in comments: I'm not a professional mathematician, and if this answer isn't the best I can to delete it, many thanks. This is the post with title Gadgets as primality tests, question asked on MathOverflow (Nov 28 '19) where were added two references in the body of my post (the reference [2] is in Spanish), and a professor added an answer. $\endgroup$ – user250478 Jun 9 at 16:17
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There are connection between some physical system and the Riemann zeta function. Here is one of the related question at mathoverflow:

https://mathoverflow.net/questions/54501/riemann-zeta-function-connection-to-quantum-mechanics/54515#54515

As I answered there the I recently found the article about this topic which may give you some answer: "Physics of the Riemann Hypothesis" http://arxiv.org/abs/1101.3116

Using the Riemann function you can build primality (at least probabilistic) test.

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