Number Theory and Theory of Equations

The theory of numbers is concerned with properties of the natural numbers (positive integers) 1, 2, 3, .... These numbers have fascinated amateur as well as professional mathematicians through the ages. A peculiarity of the subject is the great difficulty experienced in verifying results which are simple to state and which are suggested quite naturally by numerical evidence. ``It is just this," said Gauss, ``which gives the higher arithmetic [number theory] that magical charm which has made it the favoritescience of the greatest mathematicians."

One of the earliest number theory problems (going back at least 4000 years) must have been that of solving the ``Pythagorean'' equation$x^2 in integers.There is no difficulty in finding a large number of integer solutions to this equation by trial and error (some examples are $3^2). Finding all solutions requires understanding rather than patience. The problem of solving$ in integers is not very difficult and we will deal with it early in the course.

One is naturally led to ask next: Are there positive integers $x, satisfying$x^3 ? This is a much more difficult question than the previous one; the answer turns out to be ``No''. The corresponding question about$x^n, where n is an arbitrary positive integer, is one of the oldest and most famous problems in number theory. Despite recent dramatic progress on it, the problem remains unsolved. This is just one illustration of the kind of question we shall investigate.

Our study of number theory will have a decidedly algebraic flavour. It will include topics such as the algebraic properties of the integers, modular arithmetic, arithmetic in some quadratic fields, linear and quadratic equations over the integers, and continued fractions.

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The final grade may be based on class tests, assignments and a final examination.