Number theory · Maths EE idea · Ambitious
Solving Pell's equation with continued fractions
A research question to start from
How does the continued fraction expansion of √d produce the fundamental solution of x² − dy² = 1, and how large can that solution be for small d?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
Links two beautiful areas, gives you an algorithm to analyse, and produces surprising data (some fundamental solutions are enormous).
Mathematics you would need
- Simple continued fractions and convergents
- Periodicity of the expansion of √d
- Recurrence relations
- Proof by induction
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Compute continued fractions of √d for d up to 30 and find the fundamental solutions.
- Explain why the convergent at the end of the first period (or of the second, when the period is odd) gives a solution.
- Show how all solutions are generated from the fundamental one.
- Investigate how the size of the fundamental solution behaves as d varies.
Scope and difficulty
Ambitious. Ambitious: the periodicity proof is hard; it is fine to reference it and focus on why convergents work.
Pitfalls
- Too much theory, too little analysis of your own examples.
- Claiming a pattern in solution sizes without evidence.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Pell's equation continued fraction convergents; fundamental solution. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).