Roelof J. Stroeker

Elliptic binomial diophantine equations

The complete sets of solutions of the equation ( n k ) = ( m l ) are determined for the cases (k, l) = (2, 3), (2, 4), (2, 6), (2, 8), (3, 4), (3, 6), (4, 6), (4, 8). In each of these cases the equation is reduced to an elliptic equation, which is solved by using linear forms in elliptic logarithms. In all but one case this is more or less routine, but in the remaining case ((k, l) = (3, 6)) we had to devise a new variant of the method.

On a quartic diophantine equation

In this paper we consider the quartic diophantine equation 3(y2 – 1) = 2x2(x2 – 1) in integers x and y. We show that this equation does not have any other solutions (x, y) with x?0 than those given by x = 0,1,2,3,6,91. Two approaches are emphasized, one based on diophantine approximation techniques, the other depends on the structure of certain quartic number fields.

Counting and Summation

The previous chapters should have given you some idea of the possibilities of computer algebra for Analysis and Linear Algebra. In this chapter, we shall focus on another field of application. In contrast to the continuous (or limit) processes we have encountered so far, in the present chapter we shall work with finite processes such as ordinary counting. We shall have occasion to look into the ways Maple can assist us with counting and summation processes, both numerical and symbolic. The examples we have chosen are mainly taken from the fields of combinatorics and discrete probability theory.

Matrices and Vectors

Just like Mathematical Analysis, Linear Algebra too offers a number of techniques of great importance for many areas of mathematical application. We leave Analysis for the moment; in a later chapter we shall return to it with topics like differentiation and integration of mathematical functions. Although the present chapter is not of an analytic nature, what we have learned so far, especially about functions, will be rather useful to us in this chapter as well.

Vector Spaces and Linear Mappings

In this closing chapter we will be engaged again with notions and techniques typical for the field of Linear Algebra, but here our attention will also be focused on the abstract structure of vector spaces and linear mappings (or linear transformations) between them. Vectors—the elements of a vector space—do not need to be ordered lists of numbers, polynomials can also be viewed as vectors. Of course, matrices continue to play a prominent role, namely as representations of linear mappings between vector spaces with predetermined bases.

A Tour of Maple V

In this chapter you will learn by example how to get on with the Maple computer algebra system (CA system or CAS for short). In getting to know Maple, we shall discover how to deal with basic concepts and techniques such as giving instructions to the computer, correcting errors in input lines, reading input and writing output files, and we shall get acquainted with first principles of Maple’s programming language. For the time being we shall merely focus on getting familiar with the computer and the Maple CA system, and not on mathematics as yet. Later, in the chapters to follow, this practical knowledge will be supplemented in order to let Maple play an active role in helping you to improve your understanding of difficult mathematical concepts, and to generally enhance your overall knowledge of mathematics. These chapters introduce the use of the Maple system in branches of mathematics such as Mathematical Analysis, Linear Algebra, Probability Theory, and Discrete Mathematics. Also, in passing, we shall briefly pay attention to the graphical possibilities a CA system has to offer.

Functions and Sequences

After the first chapter’s tour of Maple, in which you may have gained some insight into the workings of the Maple system, it is time to concentrate on mathematics. Naturally, you won’t be an accomplished Maple user just yet, but, provided you carefully read the first chapter and worked through both worksheets, you should have some idea as to how Maple could assist in extending and deepening your knowledge of mathematics. Moreover, in this chapter and the ones to come you will gradually learn a great deal about Maple, so that at the end of this book there is a good chance that you will be a reasonably proficient user of the Maple system with an improved insight in mathematics to boot. Anyhow, that is what we aim for. But before reaching that state of affairs you need quite a bit of practice, so let us continue to explore the Maple avenues without further delay.

Derivative and Integral

After having completed a first tour of applications embracing several branches of mathematics, we turn to Mathematical Analysis again. We are now in a position to put to good use the knowledge of functions and series, and of limits and continuity, picked up in a previous chapter. In this chapter, the emphasis is on differentiation and integration of functions and the possibilities a computer algebra system provides for gaining more insight into these fundamental concepts of Analysis.