The part played by L. Euler in the development of number theory is well known: he laid the foundations of the theory of residues of powers, discovered the law of quadratic reciprocity, invented analytic methods to investigate number-theoretical questions, and so on.
Far less known are his achievements in the realm of Diophantine analysis, above all his results on the solution of Diophantine equations in rational numbers. However it is precisely Euler’s work in Diophantine analysis that represents the completion of the stage in the development of that science that might conditionallybe termed “elementary algebraic”. This stage was characterized by the use mainly of elementary algebraic means to investigate rational solutions of Diophantine equations; and within the limits of this approach important methods were developed for solving indeterminate equations of degrees two, three, and four, defining curves of genus zero or one. By “Diophantine analysis” we shall throughout mean only that portion of the subject concerned with finding rational solutions of indeterminate equations. Note also that occasionally we shall resort to using geometrical terminology – which is, of course, now standard for the subject – but this should not be taken as implying that Euler used such language.
The most important of Euler’s results in Diophantine analysis are noted in the essay of the present collection. Here we examine these results more closely, and also consider certain other little-known investigations of Euler’s.
To the solution of Diophantine equations L. Euler devoted around 40 articles and several chapters of his textbook Vollstandige Anleitung zur Algebra. There are also a great many notes on this topic among his manuscripts. The bulk of these articles are concerned with the investigation of particular Diophantine equations or systems of these. Faithful to the tradition of Diophantus and Fermat, Euler employs purely algebraic means – the apparatus of algebraic transformations, permutations, and substitutions – to obtain his solutions. In each particular case, in feeling his way towards the goal of eliciting rational solutions he shows extraordinary artistry in the application of the pertinent formulaic apparatus – in the present case algebraic, although this is characteristic of Euler’s mathematical creativity quite generally.