UNREASONABLE EFFECTIVENESS OF NUMBER THEORY 3
led to the discovery of elemental isotopes.
And finally, small divergencies in the atomic weight of pure isotopes from
exact integers constituted an early confirmation of Einstein's famous equation
long before the "mass defects" implied by these integer
discrepancies blew up into those widely noticed, infamous mushroom clouds.
On a more harmonious theme, the role of integer ratios in musical scales
has been appreciated ever since Pythagoras first pointed out their importance.
The occurrence of integers in biology - from plant morphology to the genetic
code - is pervasive. It has even been hypothesized that the North American
17-year cicada selected its life cycle because 17 is a prime number, prime
cycles offering better protection from predators than nonprime cycles. (The
suggestion that the 17-year cicada "knows" that 17 is a Fermat prime has yet
to be touted though.)
Another reason for the resurrection of the integers is the penetration of our
lives achieved by that 20th-century descendant of the abacus, the digital
computer. (Where did all the slide rules go? Ruled out of most significant
places by the ubiquitous pocket calculator, they are sliding fast into restful
An equally important reason for the recent revival of the integer is the
congruence of congruential arithmetic with numerous modern developments
in the natural sciences and digital communication - especially "secure"
communication by cryptographic systems. Last not least, the proper
protection and security of computer systems and data files against computer
"viruses" and other intrusions rest largely on keys ("digital signatures")
based on congruence relationships.
In congruential arithmetic, what counts is not a numerical value per se, but
rather its remainder or residue after division by a modulus. Similarly, in wave
interference (be it of ripples on a lake or electromagnetic fields on a hologram
plate) it is not path differences that determine the resulting interference
pattern, but rather residues after dividing by the wavelength. For perfectly
periodic events, there is no difference between a path difference of half a
wavelength and one-and-a-half wavelengths: in either case the interference
will be destructive.
One of the most dramatic consequences of congruential arithmetic is the
existence of the chemical elements as we know them. In 1913, Niels Bohr
postulated that certain integrals associated with electrons in "orbit" around
the atomic nucleus should have integer values, a requirement that 10 years
later became comprehensible as a wave interference phenomenon of the newly