Number theory · Maths EE idea · Accessible
Generating every Pythagorean triple
A research question to start from
How can every primitive Pythagorean triple be generated from two integers, and what does the parametrisation reveal about which numbers can be the hypotenuse?
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
An accessible proof that every student can follow and extend, with a natural follow-up question about the hypotenuse.
Mathematics you would need
- Greatest common divisors and coprimality
- Parity arguments
- Rational points on the unit circle
- Factorisation of integers
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
- Derive the parametrisation by factorising c² − a² and by the rational-point method on the unit circle; compare the two.
- Prove that the parametrisation gives every primitive triple exactly once.
- Characterise which integers occur as hypotenuses and test the claim computationally.
Scope and difficulty
Accessible. Accessible; reach the top bands through rigour and the comparison of two proofs.
Pitfalls
- Assuming the parametrisation covers all triples without proof.
- Mixing primitive and non-primitive triples.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Euclid's formula Pythagorean triples; rational points on the circle. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).