Fermat's Last Theorem

Fermat's Last Theorem is a famous math puzzle. It says that there are no whole numbers a, b, and c that can make the equation aⁿ + bⁿ = cⁿ true when n is bigger than 2. For n = 1 and n = 2, there are many answers, but for higher numbers, it seemed impossible.

The problem was first written down by a mathematician named Pierre de Fermat in 1637. He said he had a proof but it was too big to write in the margin of his book, Arithmetica. For over 350 years, many mathematicians tried to prove it but could not.

Finally, in 1994, a mathematician named Andrew Wiles found a proof. His work solved Fermat's Last Theorem and also helped prove other important ideas in math, like the modularity theorem. Because of this, Wiles received the Abel Prize in 2016. Fermat's Last Theorem showed how unsolved problems can help math grow and inspire new discoveries.

Overview

The Pythagorean equation (x^2 + y^2 = z^2) has many solutions in whole numbers, called Pythagorean triples, like 3, 4, and 5. In around 1637, Pierre de Fermat claimed that for any number greater than 2, the equation (a^n + b^n = c^n) has no solutions in whole numbers. Fermat said he had a proof but never wrote it down, and it took over 350 years for mathematicians to finally solve this puzzle.

Fermat’s claim became one of mathematics’ biggest unsolved problems. Over time, mathematicians proved it for certain values, but the full proof remained out of reach. It wasn’t until the work of Andrew Wiles in 1995, building on ideas from other mathematicians, that the theorem was finally proven true. His work connected Fermat’s theorem to another deep mathematical idea, showing that the two problems were linked.

Mathematical history

Pythagoras and Diophantus

Main article: Pythagorean triple

Long ago, people found that a triangle with sides 3, 4, and 5 has a right angle. This works because 3 squared plus 4 squared equals 5 squared. These special sets of numbers are called Pythagorean triples, named after the mathematician Pythagoras. Other examples are 5, 12, and 13. There are many such triples.

Main article: Diophantine equation

Fermat's equation, xⁿ + yⁿ = zⁿ, is a type of Diophantine equation, named after Diophantus. Diophantus studied these equations and made methods to solve some of them. His big book, the Arithmetica, has only partly survived. Fermat's Last Theorem came from reading this book.

Fermat's conjecture

Around 1637, a mathematician named Pierre de Fermat wrote in his book that it is impossible to split a cube into two cubes, or a fourth power into two fourth powers, or any higher power into two same powers. He said he had a great proof but there was no space to write it down. After Fermat died, his son shared this note in a new edition of the book. Even though it wasn’t proved then, this note became known as Fermat's Last Theorem.

Only one proof by Fermat has survived, for when the number n is 4. Fermat asked other mathematicians to prove the cases where n is 3 and n is 4, but he never asked about the general case. We don’t know if Fermat really had a proof for all numbers, but it seems unlikely.

Proofs for specific exponents

Main article: Proof of Fermat's Last Theorem for specific exponents

Fermat proved the case for n = 4 using a method called infinite descent. Later, other mathematicians proved it for more numbers. For example, Leonhard Euler proved the case for n = 3, but his proof had a small mistake. The case for n = 5 was proved by Legendre and Dirichlet, and the case for n = 7 was proved by Lamé.

Early modern breakthroughs

In the early 1800s, Sophie Germain found new ways to try to prove Fermat's Last Theorem for all numbers. She used special primes and showed that if certain conditions were true, these primes would affect the numbers in Fermat's equation. But she could not prove the theorem for all numbers.

Connection with elliptic curves

The proof of Fermat's Last Theorem was later connected to the Taniyama–Shimura–Weil conjecture. This conjecture says that every elliptic curve is modular. In the 1980s, mathematicians showed that this conjecture and Fermat's equation are related. If Fermat's equation had an answer, it would make an elliptic curve that was not modular, which would go against the conjecture.

Wiles's general proof

Main articles: Andrew Wiles and Wiles's proof of Fermat's Last Theorem

In 1994, mathematician Andrew Wiles proved a special part of the modularity theorem for certain elliptic curves. With earlier work by Ken Ribet, this proved Fermat's Last Theorem. Wiles's proof was published in 1995 and has helped many areas of mathematics.

The full modularity theorem was later proved by other mathematicians using Wiles's work.

Relationship to other problems and generalizations

Fermat's Last Theorem looks at solutions to the equation aⁿ + bⁿ = cⁿ where a, b, and c are positive whole numbers and n is an integer greater than 2. This theorem has inspired many similar problems and ideas.

One idea is the Beal conjecture, which asks whether there are any solutions when the exponents can be different and the numbers have no common factors. Another idea is the Fermat–Catalan conjecture, which suggests there are only a few special cases where such equations might work with different exponents. These ideas help mathematicians learn more about what kinds of numbers can fit into special equations.

Prizes and incorrect proofs

In 1816 and again in 1850, the French Academy of Sciences offered a prize for proving Fermat's Last Theorem. Later, in 1908, a German man named Paul Wolfskehl offered 100,000 gold marks to anyone who could solve it. Many people tried, but most of their proofs were wrong.

Finally, in 1997, a mathematician named Wiles solved the theorem. He received the prize money and later won the Abel Prize in 2016 for his work.

In popular culture

Fermat's Last Theorem is famous not just in math, but also in popular culture. It appeared in a 1954 short story where a mathematician makes a deal with the Devil. The theorem was also mentioned in a 1989 episode of Star Trek: The Next Generation, where Captain Picard says it is still unsolved even in the 24th century.

The book Fermat's Last Theorem by Simon Singh became a bestseller in the United Kingdom, and Singh's documentary on the topic won an award. In a 1998 episode of The Simpsons, Homer writes an equation on a blackboard that seems to solve the theorem, but it is actually wrong.