Pell equations (60 sati)
Course description
Let 𝑑 be a positive non-square integer. The equation 𝑥 2 − 𝑑𝑦 2 = 1 in integers 𝑥 and 𝑦 is
called Pell equation. It is known that this equation always always infinitely many solutions
and we furthermore know how to easily generate all solutions if we know the fundamental
one. In this course we will present the methods of finding the fundamental solution. Besides
Pell equation we will also consider its generalization. Namely, we will consider the pellian
equation 𝑥 2 − 𝑑𝑦 2 = 𝑁, where 𝑁 is non-zero integer. Even though that equation can have no
solution, if it has a solution we can generate all solutions using the fundamental solution of
associated Pell equation 𝑥 2 − 𝑑𝑦 2 = 1.
In the first part of the course we will present the theory of continued fractions and show how
we can use it in finding the fundamental solution. Unfortunately, it will be shown that this
method is inconvenient for application when integer 𝑑 is large (>1015 ).
In the second part of the course we will see that for large values of 𝑑 it is easier to compute
the regulator 𝑅 of associated real quadratic number field 𝑄(√𝑑). However, finding the
regulator can still be difficult, especially when 𝑑 is very large (>1025 ). Actually, we can
never compute the exact value of the regulator 𝑅 since it is a transcendental number, so we
are content to find a rational number 𝑅′ which is near to 𝑅. Here we will present the most well
known methods for finding the number 𝑅′ and we will show how it helps us in solving the
problem of finding the fundamental solution of the Pell equation. At the end we will show
improvements of some methods that were done in recent years and we will compare the
complexity of shown algorithms.
It will be assumed that the students are familiar with the basic notions and results from
number theory, at the level covered in the undergraduate course Number Theory.
Literature
1. H. Cohen: Number Theory. Volume I: Tools and Diophantine Equations, SpringerVerlag, Berlin, 2007.
2. H. Cohen: Number Theory. Volume II: Analytic and Modern Tools, Springer-Verlag,
Berlin, 2007.
3. M.J. Jacobson, H.C. Williams: Solving the Pell Equation, Springer, New York, 2009.
4. T. Nagell: Introduction to Number Theory, AMS Chelsea Publishing, New York,
1981.
5. I. Niven, H.S. Zuckerman, H. Montgomery, An Introduction to the Theory of
Numbers, 5th ed., John Wiley and Sons, New York, 1991.
6. N. P. Smart: The Algorithmic Resolution of Diophantine Equations, Cambridge
University Press, Cambridge, 1998.