DescriptionDiophantine equations have an immediate appeal; they are--typically very simple--equations in at least two variables with integer coefficients, for which one wants to find all solutions--if any--in integers. In contrast to their simple description they often demand subtle tools and ingenious ideas for their solution. Famous examples are1) Pell's equation: given an integer D, does have a solution in integers with non-zero y? This question leads almost inevitably to the study of continued fractions and of certain quadratic fields; 2) Fermat's Last Theorem (now Wiles's Theorem)
with an integer n>2, for the tackling of which people have developed a huge variety of tools--a prominent one being the infinite descent (some glimpses into Wiles's proof could be taken); 3) the four-square theorem: each natural number N can be written as a sum of four integer squares (summands 02 being allowed); one can in fact say in how many different ways this is possible for a given N; a related theorem states which N can be written as a sum of two integer squares; 4) Euler conjectured in the 18th century that there are no solutions in integers of the form
but only 20 years ago Elkies (and independently Zagier) have found--in fact infinitely many--counterexamples, the smallest one being given by Frye as
5) often one is also interested in studying the rational solutions of the equations in question: for example, does there exist a right triangle with rational sides and area equal to 1? Seemingly unrelated, a prize question in the Sunday Telegraph of London (1.1.1995) asked for a solution of where A, B are positive rational numbers; both questions have their natural setting in the theory of elliptic curves, where arithmetic and geometric ideas blend into each other in an intriguing way.
As there are many different levels on which diophantine equations can be studied, from the very elementary to the rather sophisticated, this project can easily be taylored to the student's background. Depending on the topic chosen the project could involve computer work, theoretical investigation or a combination of the two. PrerequisitesThere are no particular prerequisites although having already an idea about rings and polynomials as in Algebra and Number Theory II or Number Theory III may be useful for tackling deeper questions. Prior knowledge about elliptic curves would allow to take a more geometric viewpoint and to cover more demanding topics related to the ones in 5) above, or even to Wiles's work on Fermat's Last Theorem.ResourcesThe following book covers most of the above, and much more, from a classical point of view:
L. J. Mordell: Diophantine Equations or The following books give a more leisurely--and enticing--introduction to selected Diophantine problems:
K. Ireland, M. Rosen: A Classical Introduction to Modern Number Theory, Chapter 17 or
W. Scharlau, H. Opolka: From Fermat to Minkowski, Chapters 2,3,5 or
K. Kato, N. Kurokawa, T. Saito: Number Theory 1, Fermat's Dream, Chapters 0,1 or
Z. I. Borevich, I. R. Shafarevich: Number Theory, Chapter I Elliptic curves are especially nicely treated in J.T. Tate, J.H. Silverman Rational Points on Elliptic Curves (Undergraduate Texts in Mathematics Springer) Historical approaches to Fermat's last theorem can be found in P. Ribenboim 13 Lectures on Fermat's Last Theorem There are more modern books on the subject, for example Y. Hellegouarch: Invitation to the Mathematics of Fermat-Wiles and a very sophisticated one Elliptic curves, modular forms and Fermat's last theorem : proceedings of a conference held in the Institute of Mathematics of the Chinese University of Hong Kong / edited by John Coates, S.T. Yau. Useful links on the web about diophantine equations are given on (the bottom of) Dave Rusin's home page: ("Selected topics") |
email: Herbert Gangl