Real life uses of prime numbers (in physics/engineering) [closed]

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.

• 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
• This question question is gonna enter HNQ list. With 50+ upvotes. That's my prediction. – Jaideep Khare Feb 22 '18 at 20:39
• @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
• This site may list connections of interest. – Jyrki Lahtonen Feb 22 '18 at 21:51
• 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. – user253751 Feb 22 '18 at 23:39

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.

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?

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.

• That's one theory -- it's the one I've been taught in school. There is a little more information on Wikipedia. – Stefan Mesken Feb 22 '18 at 20:45
• Yes, I love this example! – Samuel Feb 22 '18 at 20:45
• Exactly what I had in mind. Also was one answer to the older question. Nevertheless a good one. – M. Winter Feb 22 '18 at 20:51

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.

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

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.

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.