Prime Numbers

8.6 Curious, anecdotal, and interdisciplinary references to primes 425
they pertain to the technology of error-correcting codes, discrete Fourier
transforms (DFTs) over fields relevant to acoustics, the use of the Möbius
µ and other functions in science, and so on. To convey a hint of how
far the interdisciplinary connections can reach, we hereby cite Schroeder’s
observation that certain astronomical experiments to verify aspects of
Einstein’s general relativity involved such weak signals that error-correcting
codes (and hence finite fields) were invoked. This kind of argument shows how
certain cultural or scientific achievements do depend, at some level, on prime
numbers. A pleasingly recreational source for interdisciplinary prime-number
investigations is [Caldwell 1999].
In biology, prime numbers appear in contexts such as the following one,
from [Yoshimura 1997]. We quote the author directly in order to show how
prime numbers can figure into a field or a culture, without much of the
standard number-theoretical language, rather with certain intuitive inferences
relied upon instead:
Periodical cicadas (Magicicada spp.) are known for their strikingly
synchronized emergence, strong site tenacity, and unusually long (17- and
13-yr) life cycles for insects. Several explanations have been proposed for
the origin and maintenance of synchronization. However, no satisfactory
explanations have been made for the origins of the prime-numbered life
cycles. I present an evolutionary hypothesis of a forced developmental delay
due to climate cooling during ice ages. Under this scenario, extremely low
adult densities, caused by their extremely 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
break down with other synchronized cycles.
It is interesting that the literature predating Yoshimura is fairly involved, with
at least three different explanations of why prime-numbered life cycles such
as 13 and 17 years would evolve. Any of the old and new theories should, of
course, exploit the fact of minimal divisors for primes, and indeed the attempts
to do this are evident in the literature (see, for example, the various review
works referenced in [Yoshimura 1997]). To convey a notion of the kind of
argument one might use for evolution of prime life cycles, imagine a predator
with a life cycle of 2 years—an even number—synchronized, of course, to the
solar-driven seasons, with periodicity of those 2 years in most every facet of
life such as reproduction and death. Because this period does not divide a 13or
17-year one, the predators will from time to time go relatively hungry. This
is not the only type of argument—for some such arguments do not involve
predation whatsoever, rather depend on the internal competition and fitness
of the prime-cycle species itself—but the lack of divisibility is always present,
as it should be, in any evolutionary argument. In a word, such lines of thought
must explain among other things why a life cycle with a substantial number
of divisors has led to extinction.