Pell Equation is an innocuous equation of degree 2 in two variables:
Since high school we know that in Cartesian coordinates, the equation has the form of a hyperbola. Starting with Pythagoras mathematicians have been interested in understanding integral solutions, i.e. solutions in which both x and y are integers. Partial solutions to the problem were known to the Greeks and the Indian school in the 7th and 8th century but only Lagrange was able to prove thet if n is not a square there are infinitely many integral solution to the Pell equation.
This apparently easy and uninteresting result puts together various aspect of Number
Theory, e.g. continued fractions, units in real quadratic number fields, rational approximations
to square roots, and many more. In this project we will discover the secrets
underlying Pell’s equation (for example that Euler attributed extra merit to Mr. Pell) as
well as many related aspects of Number Theory that will open the way to extremely interesting
topics and problems, some of which are still widely open.
Obs! För GU-studenter räknas projektet som ett projekt i Matematik (MMG900/MMG910).
Förkunskaper: Basic knowledge of linear algebra and algebraic structures is expected. In particular any
undergraduate in Mathematics and Physics has the background to enroll in this project. Willing students from other majors might be enrolled too, but a preliminary discussion with the project supervisor is strongly encouraged.
Examinator Maria Roginskaya
Institution Matematiska vetenskaper