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