Number theory and the real world

Abstract

Number theory has been considered since time immemorial to be the very paradigm of pure (some would say useless) mathematics. According to Carl Friedrich Gauss, the "Princeps Mathemat icorum", "mathematics is the queen of sc iences--and number theory is the queen of mathemat ics" . In fact, in Chinese the very name of mathematics is Number Science. What could be more beautiful than a deep, satisfying relation between whole numbers? (One is almost tempted to call them wholesome numbers.) Indeed, it is hard to come up with a more appropriate designation than their learned name: the integers, meaning the "untouched ones". How high they rank, in the realms of pure thought and aesthetics, above their lesser brethren: the real and complex numbers, whose first names virtually exude unsavory involvement with the complex realities of everyday life! Yet the theory of integers can provide totally unexpected answers to real-world problems. In fact, discrete mathematics is taking on an ever more important role. If nothing else, the advent of the digital computer and digital communication has seen to that. But even earlier, in physics, the emergence of quantum mechanics and discrete elementary particles put a premium on the methods and, indeed, the spirit of discrete mathematics. In mathematics proper, Hermann Minkowski, in the preface to his introductory book on number theory, Diophantische Approximationen, published in 1907 (the year he gave special relativity its proper four-dimensional clothing in preparation for its journey into general covariance and cosmology) expressed his conviction that the "deepest interrelationships in analysis are of an arithmetical nature". Or, on another occasion: "The primary source (Urquell) of all of mathematics are the integers". Yet much of our schooling concentrates on analysis and other branches of continuum mathematics to the virtual exclusion of number theory, group theory, combinatorics and graph theory. As an illustration, at a recent symposium on information theory, the author met several young women, formally trained in mathematics and working in the field of primality testing, who, in all their studies up to the Ph.D., had not heard a single lecture on number theory! Or, to give an earlier example, when Werner Heisenberg discovered "matrix" mechanics in 1925, he didn't know what a matrix was (Max Born had to tell him), and neither Heisenberg nor Born knew what to make of the appearance of matrices in the context of the atom. (David Hilbert is reported to have told them to go look for a differential equation with the same eigenvalues, if that would make them happier. They did not follow Hilbert's well-meant advice and thereby may have missed discovering the Schr6dinger wave equation.) Integers have repeatedly played a crucial role in the