Elementary Theory of Numbers



General Publishing Company, Ltd. | ISBN 0-486-66348-5 | ENGLISH | PDF | 136 PAGES | 7.32 MB

Introduction
1-1 What is number theory? In number theory we are concerned with properties of certain of the integers (whole numbers)
... , -3, -2, -1,0, 1, 2, 3, ... ,
or sometimes with those properties of real or complex numbers which depend rather directly on the integers. It might be thought that there is little more that can be said about such simple mathematical objects than what has already been said in elementary arithmetic, but if you stop to think for a moment, you will realize that heretofore integers have not been considered as interesting objects in their own right, but simply as useful carriers of information. After totaling a grocery bill, you are interested in the amount of money involved, and not in the number representing that amount of money. In considering sin 310 , you think either of an angular opening of a certain size, and the ratios of some lengths related to that angle, or of a certain position in a table of trigonometric functions, but not of any interesting properties that the number 31 might possess.
The attitude which will govern the treatment of integers in this text is perhaps best exemplified by a story told by G. H. Hardy, an eminent British number theorist who died in 1947. Hardy had a young protege, an Indian named Srinivasa Ramanujan, who had such a truly remarkable insight into hidden arithmetical relationships that, although he was almost uneducated mathematically, he did a great amount of first-rate original research in mathematics. Ramanujan was ill in a hospital in England, and Hardy went to visit him. When he arrived, he idly remarked that the taxi in which he had ridden had the license number 1729, which, he said, seemed to him a rather uninteresting number. Ramanujan immediately replied that, on the contrary, 1729 was singularly interesting, being the smallest positive integer expressible as a sum of two positive cubes in two different ways, namely 1729 = 103 + 93 = 123 + 13 ! It should not be inferred that one needs to know all such little facts to understand number theory, or that one
needs to be a lightning calculator; we simply wished to make the point that the question of what the smallest integer is which can be represented as a sum of cubes in two ways is of interest to a number theorist" It is interesting not so much for its own sake (after all, anyone could find the answer after a few minutes of unimaginative computation), but because it raises all sorts of further....

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