DescriptionDiophantine equations are equations with integer coefficients, for which one wants to find all solutions--if any--in integers (or in rational numbers). 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) Pythagoras's equation x2 + y2 = z2 and Fermat's equation
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; 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 neatly characterises those N which 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 though some of the things about polynomials in Algebra and Elementary Number Theory may be useful for tackling deeper questions. A good corequisite is Number Theory.ResourcesThe following book covers most of the above, and much more:L. J. Mordell: Diophantine Equations A more recent book emphasising the algorithmic aspects is
N. Smart: The algorithmic resolution of diophantine equations
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 ,
W. Scharlau, H. Opolka: From Fermat to Minkowski, Chapters 2,3,5 ,
K. Kato, N. Kurokawa, T. Saito: Number Theory 1, Fermat's Dream, Chapters 0,1 , |
email: Herbert Gangl