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

E =

mc2,

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

oblivion.)

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