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Proving Fermat’s last theorem. FLT and lessons on the nature and practice of mathematics. The theorem. There is no (non-zero) solution where x,y,z are integers and n>2 for: x n + y n = z n
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Proving Fermat’s last theorem FLT and lessons on the nature and practice of mathematics
The theorem • There is no (non-zero) solution where x,y,z are integers and n>2 for: xn + yn = zn Fermat writes in a margin that he has a marvellous proof, but there’s not enough space to present it.
It looks like it must be true, but there’s no proof. • Why does that matter?
Proving FLT • Taniyama-Shimura Conjecture • All elliptic curves can be expressed as modular functions. • It doesn’t matter for our purposes if you don’t know what that means!
Epsilon conjecture (Frey) – • If FLT is false, then there can be non-modular elliptic curves. • Bit of logic – if p, then q. q is false. So p is false (modus tollens). • So…
… if we can prove Frey and then prove the Taniyama-Shimura Conjecture… • FLT is true.
Lessons about maths • Comprehensive knowledge • Connections • Insight or intuition • Authority