Basic Concepts Of Finite Congruences

Abstract

Ever since the adveny of counting man has been toying with numbers. He soon learned that some numbers behaved differently than did others, and that different combinations of different numbers also produced different results. Addition and subtraction are the same, whereas addition and multiplication are seem to be quite different. Therefore, it was not very long after starting to count that man began to develop a number theory. At first this may have been only some deep thinking old men scratching in the sand, but soon games grew out of these early number theoretic concepts. And shortly thereafter contests between persons, cities, and countries were the rage. Of course, while this was going on some people were finding that number theory had many useful aspects, and with the development of the other sciences, number theory came into its own as a practical study. One would suspect, as is the case, that much theory about numbers was developed because of the games. And as it turned out the Arabic system lends itself very well to the theory. Among other things that have resulted from combining the Arabic numerals with number theory is the digital computer. Although number theory has many useful applications does not mean that the study of numbers has become totally utilitarian. Even today, and by all appearances for a long time to come, mathematicians, professors, students, and others will he playing with the theory of numbers. Out of the knowledge accumulated over the past 2000 years and more this paper will present a small portion of number theory; that of finite congruences. Naturally, some basic material must first be presented in order to have a foundation upon which to build. In this paper we will lay only enough groundwork to develop some basic ideas of polynomial congruences of arbitrary degree, plus a few other interesting observations. Also,we will be considering only polynomials with integral coefficients. wIt may be defined only by being described, just as we may not define, but must describe, a giant sequoia. Mathematics becomes thus independent of any other branch of human knowledge. It is autonomous and in itself must be sought its nature, its structure, its laws of being. ... It is itself a living thing, developing according to its own nature, and for its ovm ends, evolving through the centuries, yet leaving its records more imperishable than the creatures of geology.” * This is mathematics and number theory is called the ’queen of mathematics”.