INTRODUCTION
The following account consists basically of a number of loosely related investigations into finding integer solutions of equations involving integers powers of a variable-length set of unknowns. In English, or in any other spoken language, this is already a bit of a mouthful. However, in the symbolic language of mathematics, the type of equations considered are expressed easily in terms of finite sums, for example
Although I have not stated explicitly in the opening sentence, the word ‘integer’ should be read, for all intents and purposes, as ‘positive integer’ unless otherwise stated. Apart from restricting search spaces, this has had the added benefit, in certain cases, of allowing me to develop tight recursive algorithms to aid in finding particular solutions. Hopefully, I will be able to give some idea of this is in the following chapters.
I chose deliberately to leave out the adjective ‘Diophantine’ in the opening line above, for several reasons. Firstly, most of the equations I consider are open-ended in that the size of the solution set of unknowns is not fixed in advance, and secondly, there are no constants, although in certain cases at least one of the unknowns is considered only to the first power.
There are several ways of considering sums of powers. These fall into two main categories, which, in order to draw attention to their close connection with familiar problems, may be grouped under the headings ‘Euler Conjecture’ and ‘Waring Conjecture’. In the first of these, we try to find the shortest sum (i.e. smallest set) necessary for a solution to exist for particular powers. In the second, we try to find the shortest sum sufficient to guarantee that solutions exist for every situation considered. These vague definitions will be made more precise at the appropriate times.
I must stress that the types of problem considered and the particular angles of approach to them have been developed over a protracted period, and with no prior attempt to provide consistency in either notation or terminology. Consequently, familiar problems may be viewed from an unusual perspective, and some may be omitted altogether.
Chapter 1 gives a full account of the current state of play regarding investigations into Euler’s Conjecture, in particular presenting the shortest representation known of a power as the sum of lesser like powers, for powers up to 16.
Chapter 2 gives technical details on the algorithms used to provide the results in Chapter 1 and elsewhere.
Chapter 3 considers the first extension of the problem by considering positively and negatively signed powers.
Chapter 4 considers extending the problem in a different direction by representing powers as sums of different powers, introducing the idea of a smallest error or deviation from full equality.
Chapter 5 considers the combination of the two different extensions defined in the previous two chapters.
Chapter 6, as a slight departure, investigates the counting of solutions of particular equations that are known to have many solutions.
Chapter 7 is devoted to Waring’s Problem, which became a theorem rather than a conjecture only recently. It presents an introduction to the problem, including the current theoretical upper bounds for G(m) for a variety of powers m, followed by purely empirical data that suggests its own upper bounds.
Chapter 8 considers the logical extension of Waring’s problem to include positively and negatively signed powers.
A number of appendices provide detailed results, and there is a comprehensive list of references.