Mathematician Andrew Wiles finally solved Fermat’s last theorem, which had been unanswered for three and a half centuries, on June 23, 1993, in the third and final of three lectures. Both the mathematical community and the media were captivated by Wiles’ announcement.
Beyond offering a satisfying answer to an old issue, Wiles’ work represents a critical turning point in the construction of a link between two significant but superficially disparate branches of mathematics.
Making links between seemingly unrelated areas of the subject is a necessity for many of the greatest mathematical discoveries, as history has shown. Mathematicians, like the two of us, can transfer problems from one branch to another through these bridges and gain access to new resources, methods, and insights.
What is Fermat’s last theorem?
Fermat’s last theorem is similar to the Pythagorean theorem, which states that the sides of any right triangle give a solution to the equation x2 + y2 = z2.
Every differently sized triangle gives a different solution, and in fact there are infinitely many solutions where all three of x, y and z are whole numbers the smallest example is x=3, y=4 and z=5.
Fermat’s last theorem is about what happens if the exponent changes to something greater than 2. Are there whole-number solutions to x3 + y3 = z3? What if the exponent is 10, or 50, or 30 million? Or, most generally, what about any positive number bigger than 2?
Around the year 1637, Pierre de Fermat claimed that the answer was no, there are no three positive whole numbers that are a solution to xn + yn = zn for any n bigger than 2. The French mathematician scribbled this claim into the margins of his copy of a math textbook from ancient Greece, declaring that he had a marvelous proof that the margin was “too narrow to contain.”
Fermat’s purported proof was never found, and his “last theorem” from the margins, published posthumously by his son, went on to plague mathematicians for centuries.
Searching for a solution
No one was able to locate Fermat’s missing proof over the following 356 years, but no one was able to refute his theory either not even Homer Simpson. Following thousands of flawed proofs, the theorem quickly earned a reputation for being extremely challenging or perhaps impossible to establish. The theorem even earned a spot in the Guinness World Records as the “most difficult math problem.”
That is not to say that there was no progress. Fermat himself had proved it for n=3 and n=4. Many other mathematicians, including the trailblazer Sophie Germain, contributed proofs for individual values of n, inspired by Fermat’s methods.
However, mathematicians need to know that Fermat’s final theorem holds true for an infinite number of numbers, not just a few. Mathematicians wanted a proof that would work for all numbers bigger than 2 at once, but for centuries it seemed as though no such proof could be found.
However, toward the end of the 20th century, a growing body of work suggested Fermat’s last theorem should be true. At the heart of this work was something called the modularity conjecture, also known as the Taniyama-Shimura conjecture.
A bridge between two worlds
The modularity conjecture proposed a connection between two seemingly unrelated mathematical objects: elliptic curves and modular forms.
Elliptic curves are neither ellipses nor curves. They are doughnut-shaped spaces of solutions to cubic equations, like y2 = x3-3x + 1.
A function known as a modular form produces another complex number from a complex number that has two parts: a real portion and an imaginary part. These functions are unique because they are quite symmetrical, which means there are many different ways they could appear.
Although the modularity conjecture hinted otherwise, there is no reason to believe that those two ideas are connected.
Finally, a proof
The modularity conjecture doesn’t appear to say anything about equations like xn + yn = zn. But work by mathematicians in the 1980s showed a link between these new ideas and Fermat’s old theorem.
First, in 1985, Gerhard Frey realized that if Fermat was wrong and there could be a solution to xn + yn = zn for some n bigger than 2, that solution would produce a peculiar elliptic curve. Then Kenneth Ribet showed in 1986 that such a curve could not exist in a universe where the modularity conjecture was also true.
Their work implied that if mathematicians could prove the modularity conjecture, then Fermat’s last theorem had to be true. For many mathematicians, including Andrew Wiles, working on the modularity conjecture became a path to proving Fermat’s last theorem.
Wiles worked for seven years, mostly in secret, trying to prove this difficult conjecture. By 1993, he was close to having a proof of a special case of the modularity conjecture which was all he needed to prove Fermat’s last theorem.
He presented his work in a series of lectures at the Isaac Newton Institute in June 1993. Though subsequent peer review found a gap in Wiles’ proof, Wiles and his former student Richard Taylor worked for another year to fill in that gap and cement Fermat’s last theorem as a mathematical truth.
Lasting consequences
The impacts of Fermat’s last theorem and its solution continue to reverberate through the world of mathematics. In 2001, a group of researchers, including Taylor, gave a full proof of the modularity conjecture in a series of papers that were inspired by Wiles’ work. This completed bridge between elliptic curves and modular forms has been and will continue to be foundational to understanding mathematics, even beyond Fermat’s last theorem.
Wiles’ work is cited as beginning “a new era in number theory” and is central to important pieces of modern math, including a widely used encryption technique and a huge research effort known as the Langlands Program that aims to build a bridge between two fundamental areas of mathematics: algebraic number theory and harmonic analysis.
Although Wiles worked mostly in isolation, he ultimately needed help from his peers to identify and fill in the gap in his original proof. As evidenced by the extensive teamwork required to complete the modularity conjecture proof, mathematics today is becoming more and more of a collaborative undertaking. The problems are large and complex and often require a variety of expertise.
So, finally, did Fermat really have a proof of his last theorem, as he claimed? Knowing what mathematicians know now, many of us today don’t believe he did. Although Fermat was brilliant, he was sometimes wrong. Although it’s understandable to mathematicians that he thought he had a proof, it’s unlikely that his proof would hold up to modern scrutiny.
