Today, while reading about an answer related Fermat’s last theorem in Quora, I came across an interesting comment that I felt was interesting.

It goes like this:

Fermat was mistaken. He could not possibly have devised a simple proof, The proof that was finally devised relied heavily on a branch of Maths that wasnt even around at the time, known as Modular Functions.

Modular Form — from Wolfram MathWorld

There was originally an unsolved problem called the Taniyama–Shimura–Weil conjecture

Taniyama-Shimura Conjecture

which states that elliptic curves over the field of rational numbers are related to modular forms. There was also a second unsolved problem, Ribets theorem (earlier called the epsilon conjecture or ε-conjecture) which is a statement in number theory concerning properties of Galois representations associated with modular forms

Ribet’s Theorem — from Wolfram MathWorld

Frey Curve — from Wolfram MathWorld

In a nutshell:

Ribet showed that Frey curves cannot be modular, so if the Taniyama-Shimura conjecture were true, Frey curves couldn’t exist and Fermat’s last theorem would follow with b even and a=-1 (mod 4). However, Frey did not actually prove that his curve was not modular. The conjecture that Frey’s curve was not modular came to be called the “epsilon conjecture,” and was quickly proved by Ribet (Ribet’s theorem) in 1986, establishing a very close link between two mathematical structures (the Taniyama-Shimura conjecture and Fermat’s last theorem) which appeared previously to be completely unrelated.

Only by solving these two problems was it realised that it also served as a proof for Fermats Last Theorem. The proof itself is over 150 pages long and consumed seven years of Wiles’s research time.

I wish to verify the authenticity of the statement above. What is more interesting is to understand what route Fermat might have possibly taken to answer this.

According to me, new findings in any research area help in aiding the long existing attempts to solve famous unsolved questions in that area. But lack of these cannot be a primary reason for the elusion of solution so far.

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