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