Fermat's Last Theorem

Jack Dikian

April 2011

Fermat's last theorem is a theorem first proposed by Fermat in the form of a note scribbled in the margin of his copy of the ancient Greek text Arithmetica. In 1637 he famously wrote in the margin of Arithmetica that he had discovered the proof however it was too large to fit in the margin.

“I have discovered a truly remarkable proof which this margin is too small to contain”.

Fermat's Last Theorem states that xn + yn = zn has no non-zero integer solutions for x, y and z when n >2.

Despite the efforts of many mathematicians, it took another 350 years for a proof to be developed. The British mathematician Andrew Wiles spent almost 6 years developing a proof that he published in 1993. However, by mid 1993, a bombshell was dropped. Several mathematicians began finding faults in the proof when refereeing Wiles' manuscript. The faults were finally repaired by Wiles and his former student Richard Taylor in late 1994.

The question remains today, had Fermat discovered a proof in the seventieth century. The proof by Wiles and Taylor utilized mathematics not available to Fermat. Ribet's proof of the epsilon conjecture in 1986 accomplished the first half of Frey's strategy for proving Fermat's Last Theorem. Wiles set off to prove the other half.

For example,

• Taniyama-Shimura conjecture for semistable elliptic curves.
  


• Horizontal Iwasawa theory


• Euler system


• Selmer groups


• Modular elliptic curves