Pell’s Equation
Integer solutions of x² − Dy² = 1.
Key idea
Integer solutions of x² − Dy² = 1.
In depth
x² − Dy² = 1, for non-square D, always has infinitely many integer solutions. The smallest is found from the continued fraction of √D, and others are generated from it.
Worked example
x² − 2y² = 1 has (3, 2); then (17, 12), (99, 70).
Formula
Real-world uses
- Finding integer solutions for problems such as square-tile arrangements and approximations of √2
Common mistake
Looking for solutions in a table for D a perfect square, where there are only trivial ones.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Pell’s Equation?
Integer solutions of x² − Dy² = 1. It is a topic at the expert level in Discrete Math & Number Theory.
What is the formula for Pell’s Equation?
The key formula is x² − Dy² = 1.
Where is Pell’s Equation used in real life?
Real-world examples: Finding integer solutions for problems such as square-tile arrangements and approximations of √2.
What should I learn before Pell’s Equation?
Make sure you are comfortable with Pythagorean Triples Parametrisation, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Pell’s Equation?
Looking for solutions in a table for D a perfect square, where there are only trivial ones.
How can I practise Pell’s Equation?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Pell’s Equation?
Move on to Fermat’s Last Theorem next.
Related expert topics in Discrete Math & Number Theory
- Strong Induction – Assume all earlier cases to prove the next.
- Well-Ordering Principle – Every nonempty set of positive integers has a least element.
- Partial Orders – Reflexive, antisymmetric, transitive relations.
- Cardinality of Sets – Comparing sizes of infinite sets.
- Countable and Uncountable Sets – ℕ and ℚ are countable; ℝ is not.
- Cantor’s Diagonal Argument – Proof that real numbers are uncountable.