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Prime numbers (or coprimes) have few well-known uses but interesting ones.

The classical example is that prime numbers are used in asymmetric (or public key) cryptography. Prime numbers and coprimes are also used in engineering to avoid resonance and to ensure equal wear of cog wheels (by ensuring that all cogs fit in all depressions of the other wheel).

Are there other known real-world, and especially physical world or engineering, applications for prime numbers?

Existing questions on the Mathematics stackexchange forum revolve mostly around computation examples (see for example Real-world applications of prime numbers?). These questions do not list physics or engineering real-world examples of prime numbers. They concentrate mostly on computational examples.

WGroleau
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Morlock
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    Just to see if I understood you: you are interested in the use/occurrence of primes (not necessarily huge primes) in physics/engineering, not so much in computer science/programming, right? – M. Winter Feb 22 '18 at 20:32
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    Yes. What real world applications, maybe in a more physical sense, are there to primes? – Morlock Feb 22 '18 at 20:35
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    This is only tangentially related but fun fact: During the recording of the 'stomp-stomp-clap' section in a church in North London of Queen's 'We Will Rock You' they kept prime unit distances from the microphone. – Stefan Mesken Feb 22 '18 at 20:39
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    This question question is gonna enter HNQ list. With 50+ upvotes. That's my prediction. – Jaideep Khare Feb 22 '18 at 20:39
  • @StefanMesken You mean people where at a coprime set of distances, or everybody was at meters from the microphone? – Morlock Feb 22 '18 at 20:40
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    No necessarily prime number $\cdot $ meters but prime number $\cdot$ some fixed unit distance (in fact prime number $\cdot$ half the distance to the first bench IIRC). Yes. – Stefan Mesken Feb 22 '18 at 20:42
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    @StefanMesken yes, you are right https://www.npr.org/2010/08/03/128935865/queens-brian-may-rocks-out-to-physics-photography – john doe Feb 22 '18 at 20:42
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    Possible duplicate of Real-world applications of prime numbers? – Andreas Feb 22 '18 at 20:46
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    @AndreasAlmgren He even mentioned this question himself and made clear why it does not match his question. The answers over there are mostly about computer science. I clarified with my first comment that this is not his intention. – M. Winter Feb 22 '18 at 20:49
  • @M.Winter, I was mostly thinking on old answer subjects reappearing. But ok! – Andreas Feb 22 '18 at 20:53
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    @AndreasAlmgren Maybe one can find a better title to set this question apart from similar ones, e.g. "Real life uses of prime numbers in Physics/Engineering". – M. Winter Feb 22 '18 at 20:55
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    @Peter It seems, from my searching, that phenomenons that have to do with a frequency or periodicity may lead engineers to use prime numbers to avoid synchronicity or resonance. I'm interested to see what other applications there may be. Apparently, they are few but do exist. – Morlock Feb 22 '18 at 21:15
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    @Peter You don't seem to have an answer, maybe this question is therefore also interesting for you. As a mathematician you should know that just because you do not see a way something can happen does not mean there is no way. – M. Winter Feb 22 '18 at 21:15
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    This site may list connections of interest. – Jyrki Lahtonen Feb 22 '18 at 21:51
  • @Peter There are some surprises here :-) – Jyrki Lahtonen Feb 22 '18 at 21:57
  • @JyrkiLahtonen Thanks, I had not come across that website. – Morlock Feb 22 '18 at 22:15
  • (1/2) Fermat primes are related to geometry, see this Wikipedia. My belief (that is a speculation) is that the so-called Mersenne primes are also related to geometry (what?, my speculation is that is related to the nature of the light-gravity). I think that primes with form like Mersenne, Germain or other constellations should have a physical meaning. The distribution of all primes is the axis Euclides(there are infinitely many)-Gauss(prime number theorem)-Riemann(the best error term via Riemann Hypothesis) –  Feb 22 '18 at 22:47
  • (2/2) Our logarithm for real/rational numbers $x>1$ is the interpolation of $\log n=\sum_{d\mid n}\Lambda(d)$ involving the von Mangoldt function. A prime is like as a fruit box without subdivisions (with the exception of the unitary subdivision). The last prime that we know typically is a Mersenne prime. The other primes, big primes, are used to send information (with safety). The problem related to perfect numbers is maybe the more ancient problem, born together the Euclidean geometry. Open problems related to prime numbers are not presented as great obstacles to the progress of science. –  Feb 22 '18 at 23:01
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    @user243301 Even if RSA is safe (which is debateable) , I still do not see what this has to do with physics. – Peter Feb 22 '18 at 23:12
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    My apologizes @Peter my ideas are speculations. –  Feb 22 '18 at 23:14
  • @JaideepKhare Your prediction seems to become true. – Peter Feb 22 '18 at 23:14
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    No source, but IIRC, when making twisted pair cable (such as standard cat6) they use coprime numbers of twists per meter on each pair to reduce interference between the pairs. – Stack Exchange Broke The Law Feb 22 '18 at 23:39
  • Do you really mean "uses"? Or do you mean "natural occurences"? – paul garrett Feb 22 '18 at 23:44
  • I mean uses in physics or engineering. – Morlock Feb 23 '18 at 00:15
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    There cannot be a single adequate answer to my question. I always find that such questions are highly interesting and disagree that they diminish the quality of stackexchange sites. I think this perspective is bad and that open questions like these are a good way for people to get interested in mathematics that would not normally read about math. Feel free to keep this on hold indefinitely of delete. – Morlock Feb 23 '18 at 13:49
  • @paulgarrett Perhaps you are interested in a prime-finding-project. If yes, look here : https://math.stackexchange.com/questions/2635516/numbers-n-of-the-form-10m2k%e2%88%9212k-1%e2%88%921-where-m-is-the-number-of/2636195?noredirect=1#comment5494309_2636195 – Peter Feb 23 '18 at 14:30
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    Does anyone use CDs anymore? I suspect there is still a lot of music files out there at $44.1$ kHz. Notice that $44100 = 2^2 \times 3^2 \times 5^2 \times 7^2$. By contrast $48000$ is not divisible by $7$. – Bill Thomas Feb 23 '18 at 21:59
  • @WGroleau primes are in the atoms...Riemann Zeta function seems to have connections with quantum-states...as primes are the builind blocks of all numbers, atoms are the building blocks of the matter... – Enzo Creti Feb 27 '18 at 09:33
  • @EnzoCreti: I think you picked the wrong name. I made no comment here and I also don't understand your response to the comment I didn't make. – WGroleau Feb 27 '18 at 11:52
  • @immibis I wish there was a source, because I'd love to know why "per meter" which is not clear to me at all! – Charles May 23 '18 at 16:29
  • @Charles probably because that's how they specify the number of twists, so it's convenient for them. – Stack Exchange Broke The Law May 23 '18 at 23:00
  • @immibis Yes, but Nature doesn’t know about meters, and the number of twists won’t be coprime in other measurements presumably. I just think it would make interesting reading since it deals with continuous vs discrete in a natural setting with some perhaps nonobvious results. – Charles May 26 '18 at 04:51
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    I think this is a real useful question. It helps people make an informed decision on whether to study prime numbers. It was clear enough to me what the author was asking. I don't think there's another Stack Exchange question like it. I think it was written pretty much the best way it can be written. I have an answer to it which is that they're useful for computing logs of positive integers. For example, once you've computed the log of every prime number up to a given number, you can just simply use addition to compute the log of every other positive integer up to that number. – Timothy Oct 09 '19 at 22:41

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Clock making is a great example. Need a movement that moves at 23/83 ticks per second, anyone? In the olden-days, we'd approximate such a fraction using what's called the Stern-Brocot tree of rational numbers, which produces an ordered set of coprime ratios of integers which spans the rationals. See here for more.

Samuel
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There are prime numbers in the biological world ... I assume for good evolutionary reasons.

For example, the Cicadas emerge every 13 or 17 years. Maybe this is to minimize the overlap with other species that also emerge only in certain years?

Here is what the Wikipedia article says about this:

The emergence period of large prime numbers (13 and 17 years) was hypothesized to be a predator avoidance strategy adopted to eliminate the possibility of potential predators receiving periodic population boosts by synchronizing their own generations to divisors of the cicada emergence period. Another viewpoint holds that the prime-numbered developmental times represent an adaptation to prevent hybridization between broods with different cycles during a period of heavy selection pressure brought on by isolated and lowered populations during Pleistocene glacial stadia, and that predator satiation is a short-term maintenance strategy.

Bram28
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Prime numbers of turbine, fan and stator blades are very common in gas turbines (airplane engines), to push the fundamental frequency of air pulses through the wake of the stators away from each other. Of course, they need to be relatively prime - if you have seventeen blades behind seventeen stators, the whole disk is experiencing a pressure pulse every 1/17th of a rotation.

Whether you need to do this depends a bit on how highly loaded your components are and your other design decisions. This paper (end of p244, start of p245) talks about this.

Atcrank
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The NTSC timing frequencies are composed of small primes so that the color subcarrier didn't beat with the horizontal line frequency by being relatively prime. Part of the constraints on the choice was that the FM sound carrier frequency had a very tight tolerance and could not be changed and still maintain backward compatibility with black and white TVs. The horizontal line frequency could be altered however.

See:

Abrahams, I. C., "Choice of Chrominance Subcarrier Frequency in the NTSC Standards", Proceedings of the I-R-E, January 1954, pp 79-80

Abrahams, I. C., "The 'Frequency Interleaving' Principle in the NTSC Standards", Proceedings of the I-R-E, January 1954, pp 81-83

Andy Walls
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One area that has interested me for some time is classical cryptography. Recently, I have become more interested in the Enigma machine. A quick lookup shows that prime and co-prime numbers may have been used deliberately to make the Enigma machine's encrypting less predictable.

From http://www.cryptomuseum.com/crypto/enigma/g/a28.htm :

The cog-wheels of all wheels are coupled via a spindle at the rear. This spindle has three small cog-wheels with a series of alternating long and short teeth. The coupling can be engaged and disengaged by using the EIN/AUS coupling lever at the top left of the machine.

In order to increase the cipher period of the machine, each wheel has a different number of notches, all being relative primes of 26. Furthermore, there is no common factor between the numbers. Wheels I, II and III have 17, 15 and 11 notches respectively.

Community
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Morlock
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Note: I've worded this very badly, please read the source instead of my question if you do not want to read what I wrote.

Prime numbers would be useful in the future (who knows when) to communicate to extraterrestrial creatures.

Any "random" place in space would have no feature or characteristics which would separate it from any other "random" place in space.

A quest for extraterrestrial life could involve searching for irregular sequences and different background noises in space than the background noise generally found.

If so, a sequence noises which could be translated into prime numbers coming from one part of the universe could most likely mean that there was and maybe is extraterrestrial life (especially intelligent extraterrestrial life) in that part of the universe.

Source: https://www.math.dartmouth.edu/~carlp/PDF/extraterrestrial.pdf

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    In the case extraterrestrial (intelligent) life exists it is quite speculative to assume that it comprehends what prime numbers are and that it even deals with math at all. It is very likely that we won't be able to communicate with extraterrestrials. They probably will be too different from us. – Peter Feb 22 '18 at 23:23