Math 126 Number Theory

Short Description
Mathematics is really very well-. connected. A topic like this, Pell’s equation, which. seems to belong to number theory, has strong con- …

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Content
Math 126 Number Theory
Prof. D. Joyce, Clark University
24 Apr 2006
Last time. Introduction to Pell equations. We
saw some examples of how to find solutions to Pell
equations using continued fractions.
Today. Theory of Pell equations. We’ll see some
theorems that show the methods work that we
looked at last time.
We’ll start with a theorem that shows how to
find infinitely many solutions if you just have one
to start with. Here’s the statement of the theorem,
followed by a lemma that we’ll use to prove it, then
we’ll have the proof of the lemma and the theorem.
Theorem. If the equation x2 . dy2 = 1 has one
solution, then it has infinitely many solutions, while
if the equation x2.dy2 = .1 has one solution, then
not only does it have infinitely many solutions but
also x2 . dy2 = 1 has infinitely many solutions.
Lemma. If (a, b) is a solution to the Diophantine
equation
x2 . dy2 = c,
then solutions to the equations
x2 . dy2 = cn
are recursively defined by
x1 = a
y1 = b
xn+1 = axn + dbyn
yn+1 = bxn +…