Fermat's Last Theorem

Podcast Transcript

For more than 350 years, a single problem stumped the world of mathematics.

The problem was extremely simple to state, yet it proved fiendishly difficult to prove.

Bounties were placed on finding a solution, and yet many failed to prove it.

Finally, in 1994, seemingly out of nowhere, a proof was offered, but it was far cry from the initial promise of being simple.

Learn more about Fermat’s Last Theorem and its legacy in the world of mathematics on this episode of Everything Everywhere Daily.

I’ve done many episodes on mathematical subjects, and I actually like doing mathematical episodes.

However, there is a problem with episodes about mathematics. There are some topics that are very difficult to do in an audio format. Most topics require a graph or at least an equation to illustrate what is being discussed.

Moreover, there is a large gap in the knowledge of mathematics among the listeners of this podcast. Some listeners consider themselves to be horrible at math and have never studied it beyond high school, and I also know there are professors of mathematics who listen to the show.

The result is that there are some topics I’d love to do episodes on, but I really can’t because I don’t think that this is a good platform.

For example, in the year 2000, the Clay Mathematics Institute published seven unsolved problems in mathematics. For each solution, they offered a one-million-dollar prize to the person who could solve it.

I’ve been thinking about an episode on this topic for over four years, and I’m not sure I could possibly do it justice in the limitations of this podcast format. I’m not sure how to explain many of the problems, and to be honest, despite having a degree in the subject, there are some of the problems I do not understand completely.

Many of the great unsolved problems today are in obscure branches of mathematics and are difficult to comprehend if you don’t specialize in that field.

The subject of this episode is much more approachable. It harkens back to a time when mathematics was becoming formalized, and it became one of the earliest and most famous unsolvable problems.

It starts with Pierre de Fermat.

Fermat, born in 1607, was a French mathematician, lawyer, and government official widely regarded as one of the founders of modern number theory.

Born in Beaumont-de-Lomagne, France, Fermat worked as a lawyer and magistrate but pursued mathematics as a passionate hobby. His contributions to mathematics include the development of analytic geometry alongside René Descartes, early work on calculus, and foundational contributions to probability theory with Blaise Pascal.

Fermat is most famous for his work in number theory, particularly the subject of this episode, what has become known as Fermat’s Last Theorem.

The origin of the theorem goes back to one of the oldest theorems in mathematics, the Pythagorean Theorem.

The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse, aka the side opposite the right angle, is equal to the sum of the squares of the other two sides.

Expressed algebraically, it would be a²+b²=c².

An example of this would be three squared plus four squared equals five squared, or 9 plus 16 equals 25.

In this case, three, four, and five are known as Pythagorean triples. It turns out that there are an infinite number of Pythagorean triples where a²+b²=c².

5, 12, and 13 are another set of Pythagorean triples. Five squared plus 12 squared equals 13 squared, or 25 plus 144 equals 169.

The Pythagorean Theorem is part of a more generalized type of equation known as a Diophantine equation.

They, too, have an ancient origin. They are named after Diophantus of Alexandria, who wrote the early mathematics text titled Arithmetica, which explores solving algebraic equations, particularly Diophantine equations, focusing on finding rational or integer solutions.

In the Arithmetica, Diophantus talks about how a square can be split into two other squares, as in the Pythagorean Theorem.

In Fermat’s personal copy of the Arithmetica, he made a short observation in the book’s margin, written in Latin.

The translation of what he wrote is as follows:

It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

Basically, Fermat claimed that instead of raising numbers to the power of two like in the Pythagorean Theorem, if you raised numbers to the power of three, four, or any other number, it would be impossible.

Moreover, and what turned this simple claim into something legendary, he said that he had found a way to prove it.

Fermat died in 1665, and five years after his death, his son published a new edition of Arithmetica, which included his father’s notes.

It was this copy of Arithmetica with Fermat’s notes, which is how Fermat’s Last Theorem came to the attention of the world, and why it was named his last theorem….because it was released after his death.

The simplicity of the theorem’s statement, combined with the mystery of Fermat’s “missing proof,” attracted mathematicians for centuries.

Fermat himself actually took the first step in attempting to solve this problem. He proved that the theorem was true for any integers raised to the power of four.

One consequence of Fermat’s proof was that it proved the theorem for every even exponent. This whittled the problem down to just proving it true for all odd prime numbers.

The next major step took place almost a century later with the great Swiss mathematician Leonhard Euler.

Euler proved that the theorem was true for all values of three. Technically, his proof contained an error that was later corrected by others.

Again, it wasn’t a general solution to the problem, but if the general theorem was true, it had to be true for three.

The biggest advance in tackling the problem was made in the early 19th century by the French mathematician Sophie Germain.

Germain approached the problem by introducing what is now referred to as Germain’s Theorem. She demonstrated that if p is an odd prime and 2p+1 is also prime, there are no solutions for such numbers.

This was a huge step beyond what Euler proved because it covered an infinite number of cases.

For example, her proof covered Euler’s proof for three because 3(2) + 1 is 7, which is prime. It also proved 5 and 11 for the same reasons.

However, it didn’t provide a proof for 7, for example.

Germain’s work was groundbreaking and earned the respect of the leading mathematicians of the day, such as Carl Friedrich Gauss. However, in her correspondence with Gauss, she used a male pseudonym because she didn’t think she’d be taken seriously as a woman at the time.

In the late 19th century, German mathematician Paul Wolfskehl left a substantial prize for the first correct proof of Fermat’s Last Theorem. This inspired a flood of submissions, all of which were invalid. Wolfskehl himself had become interested in the theorem after thinking he had found a proof but later realizing his error.

Thousands of incorrect proofs were submitted attempting to claim the prize.

What really changed were modern advancements in number theory, which were developed in the 20th century. One of the developments was Elliptic Curves and the other was modular forms.

In the 1950s, a conjecture known as the Taniyama-Shimura-Weil Conjecture was proposed. This conjecture suggested a deep connection between elliptic curves and modular forms. Though seemingly unrelated to Fermat’s theorem, it became central to its proof.

That was because, in 1986, Kenneth Ribet of the University of California at Berkeley proved that if the Taniyama-Shimura-Weil conjecture was true, then Fermat’s Last Theorem must also be true.

This shifted the problem to proving the Taniyama-Shimura-Weil conjecture.

Enter into the picture, Andrew Wiles.

Wiles was a British mathematician who developed a passion for mathematics at a young age and was inspired by Fermat’s enigmatic theorem. He earned his Ph.D. at Clare College, Cambridge, and became a leading figure in number theory, specializing in elliptic curves and modular forms.

He was working at Princeton when he read Kenneth Ribet’s work and set out to prove the Taniyama-Shimura-Weil conjecture as a means of proving Fermat’s Last Theorem.

Wiles worked by himself for years on the problem. In fact, he never told anyone that he was working on the problem.

One of the reasons he didn’t work with anyone else and never told anyone about it was because of the mystique that Fermat’s Last Thorem held.

There had been numerous amateur mathematicians who had claimed to have solved the theorem, but actually didn’t really understand it.

Some conspiracy theorists claimed that Fermat’s original “marvelous proof” existed but had been lost or suppressed the powers that be. These theories often involved no real mathematics and were purely speculative.

Crank mathematicians sometimes resubmitted their “proofs” repeatedly, even after experts refuted them. The internet era amplified this, with forums and blogs hosting numerous amateur claims.

According to mathematical historian Howard Eves, “Fermat’s Last Theorem has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published.”

Finally, after years of isolated work, in June 1993, Wiles announced his findings at a conference.

He had done it using modern techniques and proving the Taniyama-Shimura-Weil conjecture, which Kenneth Ribet showed would then prove Fermat’s Last Theorem.

However, there was something about Wiles’s proof that that most people didn’t expect.

When Fermat wrote in the margins of his book, he seemed to imply that his proof was short and elegant.

Weils proof was not. It was over a hundred pages long, and it involved a host of techniques used in modern number theory.

The proof was extremely complicated and it wasn’t something that most mathematicians could quickly verify.

Once mathematicians were able to take a look at his proof, by August several of them found an error.

He went back to the drawing board and went to work trying to fix the error in his proof.

By September 1994, he was almost ready to give up, but he finally made a breakthrough and on October 24, he submitted two papers, one which was his main proof and a second which explained his corrected of his original proof.

They were formally published in 1995.

After 358 years, Fermat’s Last Theorem was proven and was verified by the mathematical community.

Andrew Wiles work has had enormous implications beyond just solving this theorem. His work has helped advance many other related fields of mathematics.

For his work, Wiles was made a fellow of the Royal Society, was knighted, and in 2016 was the awarded the Able Prize, the equivalent of the Nobel Prize in mathematics.

He was also given a special award by the International Mathematical Union because he was ineligible to receive a Fields Medal, which was one of the most prestigious prizes in mathematics. The Fields Medal is only awarded to mathematicians under the age of 40. Wiles made his breakthrough at the age of 41.

Fermat’s Last Theorem was one of the longest unsolved problems in the history of mathematics.

For several centuries there were those who thought that Fermat’s Last Theorem was impossible to solve. There was even an episode of Star Trek: The Next Generation recorded five years before Wiles proof which had it still unsolved in the 24th century.

When it was finally solved, its solution was a form that no one expected.

Ultimately, Andrew Wiles’ 1994 proof succeeded because it built on centuries of mathematical progress, demonstrating that solving such a deep problem required modern tools and techniques far beyond what Fermat or his contemporaries had available, as well as an incredible amount of tenacity.