Prime Numbers

430 Chapter 8 THE UBIQUITY OF PRIME NUMBERS
in some asymptotic sense, Wolf finds it to hold over a very wide range of
M,N. For example, M =2 16 ,N =2 38 gives a compelling and straight line
on a (ln |Xk| 2 , ln k) plot with slope ≈−1.64. Whether or not there will be
a coherent theory of this exponent law (after all, it could be an empirical
accident that has no real meaning for very large primes), the attractive idea
here is to connect the behavior of complex systems with that of the prime
numbers (see Exercise 8.33).
As for cultural (nonscientific, if you will) connections, there exist many
references to the importance of very small primes such as 2, 3, 5, 7; such
references ranging from the biblical to modern, satirical treatments. As just
one of myriad examples of the latter type of writing, there is the piece in
[Paulos 1995], from Forbes financial magazine, called “High 5 Jive,” being
about the number 5, humorously laying out misconceptions that can be traced
to the fact of five fingers on one hand. The number 7 also receives a great
deal of airplay, as it were. In a piece by [Stuart 1996] in, of all things, a
medical journal, the “magic of seven” is touted; for example, “The seven ages
of man, the seven seas, the seven deadly sins, the seven league boot, seventh
heaven, the seven wonders of the world, the seven pillars of wisdom, Snow
White and the seven dwarves, 7-Up ... .” The author goes on to describe
how the Hippocratic healing tradition has for eons embraced the number 7
as important, e.g., in the number of days to bathe in certain waters to regain
good health. It is of interest that the very small primes have, over thousands
of years, provided fascination and mystique to all peoples, regardless of their
mathematical persuasions. Of course, much the same thing could be said about
certain small composites, like 6, 12. However, it would be interesting to know
once and for all whether fascination with primes per se has occurred over the
millennia because the primes are so dense in low-lying regions, or because the
general population has an intuitive understanding of the special stature of the
primes, thus prompting the human imagination to seize upon such numbers.
And there are numerous references to prime numbers in music theory and
musicology, sometimes involving somewhat larger primes. For example, from
the article [Warren 1995] we read:
Sets of 12 pitches are generated from a sequence of five consecutive prime
numbers, each of which is multiplied by each of the three largest numbers
in the sequence. Twelve scales are created in this manner, using the
prime sequences up to the set (37, 41, 43, 47, 53). These scales give
rise to pleasing dissonances that are exploited in compositions assisted
by computer programs as well as in live keyboard improvisations.
And here is the abstract of a paper concerning musical correlations
between primes and Fibonacci numbers [Dudon 1987] (note that the mention
below of Fibonacci numbers is not the standard one, but closely related to it):
The Golden scale is a unique unequal temperament based on the Golden
number. The equal temperaments most used, 5, 7, 12, 19, 31, 50, etc., are
crystallizations through the numbers of the Fibonacci series, of the same