Fermat's Last Theorem for n = 4: a maths EPQ idea
A title to start from
Why can Fermat's Last Theorem be proved for n = 4 with school mathematics when the general case took over 350 years?
Why it works as an EPQ
The n = 4 case is a complete, elementary proof by infinite descent, and the contrast with the general case gives you a real question to evaluate.
Scope and difficulty
Ambitious. Ambitious but bounded: prove n = 4 fully, explain the history of the general case in words without attempting it.
The maths
Builds on these A Level topics: Proof · Algebra and functions.
You would learn:
- Parametrising Pythagorean triples
- Infinite descent
- Coprimality arguments
One possible plan
- Derive the formula for primitive Pythagorean triples.
- Use it to prove there are no positive integer solutions of x⁴ + y⁴ = z², and deduce the n = 4 case.
- Explain why the method does not extend, using the history of n = 3 and the eventual proof.
- Evaluate what 'school mathematics' can and cannot do here.
Pitfalls
- Attempting to explain Wiles's proof in detail.
- Skipping the coprime cases in the descent.
Where to start reading
- Fermat's Last Theorem (Simon Singh)
- Search for: infinite descent x^4 + y^4 = z^2 proof
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