The following message was posted by Ernst Mayer on the "primes-l" mailing list (23 Jan 2000):

Doug Sloan <sloan@indy.net> wrote:

>I am trying to develop a lesson plan that includes a section on how are
>primes used. I am looking for the more down-to-earth and mundane uses or
>easily explained occurrences in nature.

I have several additional examples, some from nature and some from mathematics itself (i.e. surprising appearances of primes in other mathematical contexts):

0) Primes in geometrical construction: I quote from p.33 of a draft manuscript of the soon-to-appear book Prime Numbers: A Computational Perspective by Richard Crandall and Carl Pomerance (my annotations in []; ** indicates italicized expressions):

A very remarkable theorem on prime Fermat numbers [F_n = 2^(2^n)+1, of which F_0 - F_4 are prime, but all others whose status is known have proved composite] was proved by Gauss, again from his teen years. He showed that a regular polygon with n sides is constructible with straight edge and compass if and only if the largest odd divisor of n is a product of distinct Fermat primes. If F_0, ..., F_4 turn out to be the *only* Fermat primes, then the only n-gons that are constructible are those with n = 2^a x m with m|2^32 - 1 (since the product of these five Fermat primes is 2^32 - 1).
(Many of the remaining examples are also in the Crandall/Pomerance book.)

1) Evolutionary biology: the striking existence of two closely related species of cicadas with 13 and 17-year life cycles may be an example of adaptation driven by the special properties of prime numbers. Yoshimura (1997) suggests that (my annotations in [])

"...extremely low adult densities, caused by their exremely long juvenile stages, selected for synchronized emergence and site tenacity because of limited mating opportunities. The prime numbers (13 and 17) were selected for as life cycles because these cycles were least likely to coemerge, hybridize, and [as a consequence of hybridization] break down with other synchronized cycles."
2) Molecular biology: Yan et al. (1991) suggest that certain amino acid sequences found in genetic material exhibit patterns similar to those of binary representations of prime numbers. There may be deep connections between primes and genetics in the sense that via mutation, recombination and natural selection, genetic material evolves toward some kind of optimal balance of coding efficiency (which one might attempt to relate to the fundamental theorem of arithmetic or the Goldbach conjecture, but this is very speculative) and robustness, i.e. containing enough redundancy to permit a reasonable amount of error correction (see also item 3).

3) Error-correcting codes: the technology of such codes uses the mathematics of finite fields, e.g. arithmetic modulo some integer n the ("characteristic" of the field, which must be prime for the field property to hold. In science, anytime one deals with a very weak signal one needs error-correcting codes. For example, some of the astronomical experiments used to verify Einstein's general relativity theory involve weak signals, and require error-correcting codes (and hence prime numbers, via finite fields) to be used. Encoding schemes which permit error correction with a minimal amount of redundancy are also of fundamental importance in the emerging science of quantum computation.

4) Discrete fourier transforms (DFTs) are ubiquitous in signal processing (power spectral analysis and filtering) and statistics (convolutions and correlations). While DFTs over the complex numbers are most common in engineering and the applied sciences, one can also define analogous kinds of transforms over finite fields posessing certain desired properties. These exhibit many nice computational properties when the length of the discrete signal vectors in question is a product of distinct primes, i.e. when all the radices used in the transform are coprime.

5) In physics, certain matrices appearing in the study of chaotic quantum systems appear to have an eigenvalue spectrum closely related to (in the the limit of infinite matrix size, perhaps identical to) the complex zeros of the famous Riemann Zeta function, i.e. closely connected to the prime numbers.

A good reference on connections between applied sciences and number theory is M. R. Schroeder (1997).

Personally, for all these interesting examples, I believe there are many more examples of practical importance wherein 2 or more numbers are relatively prime (share no common divisors > 1) than there are for actual primes. Items 1, 2 and 4 above in fact rely on this rather than on strict primality of the relevant parameters. I believe orbital resonances in celestial mechanics involve ratios of coprime numbers, but you should check this to see if it is fundamental and interesting, or simply a consequence of the fact that any rational number may be expressed as a ratio of two coprime integers.

Cheers,
-Ernst

 
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