Fermat’s Last Theorem – How It Was Solved (Solution)

Fermat’s Last Theorem, a famous problem in the history of mathematics, was conjectured by Pierre de Fermat in 1637.

It states that there are no three positive integers $a$ , $b$ , and $c$ that can satisfy the equation $a^^{n} + b^^{n} = c^^{n}$ for any integer value of $n$ greater than 2.

For centuries, Fermat’s Last Theorem was one of the most notorious unsolved problems in mathematics.

Fermat himself claimed to have discovered a proof that was too large to fit in the margin of the book where he wrote the theorem.

Despite this claim, no proof by Fermat was ever found, and the theorem remained unproven for over 350 years.

Table of Contents

Andrew Wiles

The proof of Fermat’s Last Theorem was finally completed by British mathematician Andrew Wiles, with crucial assistance from Richard Taylor, in the 1990s.

Wiles’s approach to solving the theorem was to link it to a completely different area of mathematics, the modularity theorem for elliptic curves, which was conjectured at the time.

The modularity theorem (formerly known as the Taniyama-Shimura-Weil conjecture) posits a deep connection between elliptic curves and modular forms.

Wiles’s Proof for Fermat’s Last Theorem

The outline of Wiles’s proof is as follows:

Elliptic Curves and Modular Forms

Wiles showed that a certain kind of elliptic curve is modular.

This means that the elliptic curve can be associated with a modular form.

The connection between elliptic curves and modular forms was central to the strategy for proving Fermat’s Last Theorem because of the existing conjecture (Taniyama-Shimura-Weil) that suggested all elliptic curves are modular.

Linking to Fermat’s Last Theorem

Wiles utilized this connection to Fermat’s Last Theorem through the notion of a “Frey curve.”

A hypothetical solution to Fermat’s equation $a^^{n} + b^^{n} = c^^{n}$ (for $n > 2$ ) would lead to the construction of a particular elliptic curve (now known as a Frey curve).

The properties of this curve would violate the modularity theorem if such a solution existed.

Proof by Contradiction

By demonstrating that all semistable elliptic curves are modular (a key part of the modularity theorem), Wiles showed that the Frey curve could not exist.

This implies that there are no solutions to Fermat’s Last Theorem for $n > 2$ , thus proving the theorem by contradiction.

Wiles’s initial proof, presented in 1993, contained a gap.

However, over the subsequent months, with the help of Richard Taylor, Wiles was able to fix the gap.

The corrected proof was published in 1995.

Conclusion

This monumental achievement not only solved Fermat’s Last Theorem but also had profound implications for number theory, leading to significant developments in the understanding of elliptic curves and modular forms.

Wiles’s work was recognized with numerous awards, including a special recognition from the Abel Prize committee.