Fermat's Last Theorem

Fermat's Last Theorem is one of the most famous puzzles in math. 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 solutions, 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, the best mathematicians tried and failed to prove it.

Finally, in 1994, a mathematician named Andrew Wiles found a proof. His work not only solved Fermat's Last Theorem but also helped prove other big ideas in math, like the modularity theorem. Because of this achievement, Wiles received the Abel Prize in 2016. Fermat's Last Theorem showed how important unsolved problems can push math forward 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

In ancient times, it was known that a triangle with sides in the ratio 3:4:5 has a right angle. This is because, when you square two sides (3² + 4² = 9 + 16 = 25) and it equals the square of the third side (5² = 25), the triangle must have a right angle. Such triples of numbers are called Pythagorean triples, named after the ancient Greek mathematician Pythagoras. Examples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples.

Main article: Diophantine equation

Fermat's equation, xⁿ + yⁿ = zⁿ, is a type of Diophantine equation, named after the mathematician Diophantus. Diophantus studied these equations and developed methods to solve some of them. His major work is the Arithmetica, of which only a portion has survived. Fermat's Last Theorem was inspired by reading this book.

Fermat's conjecture

Around 1637, the mathematician Pierre de Fermat wrote in the margin of his copy of the Arithmetica that it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or any higher power into two similar powers. He claimed to have a marvelous proof but the margin was too small to contain it. After Fermat's death, his son produced a new edition of the book with his father's comments. Although not a proven theorem at the time, this marginal note became known as Fermat's Last Theorem.

Only one related proof by Fermat has survived, for the case where the exponent n is 4. Fermat posed the cases of n = 3 and n = 4 as challenges to other mathematicians but never posed the general case. It is uncertain whether Fermat had a valid proof for all exponents, 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 technique called infinite descent. Later mathematicians developed proofs for other specific exponents. For example, the case for n = 3 was first proved by Leonhard Euler, though his proof had a gap. 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 19th century, Sophie Germain developed new approaches to prove Fermat's Last Theorem for all exponents. She defined auxiliary primes and showed that if certain conditions were met, these primes would divide the product of the numbers in Fermat's equation. However, she did not succeed in proving the theorem for all exponents.

Connection with elliptic curves

The proof of Fermat's Last Theorem was eventually linked to the Taniyama–Shimura–Weil conjecture, which states that every elliptic curve is modular. In the 1980s, mathematicians linked this conjecture to Fermat's equation. If Fermat's equation had a solution, it would create an elliptic curve that was not modular, contradicting the conjecture.

Wiles's general proof

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

In 1994, mathematician Andrew Wiles succeeded in proving a special case of the modularity theorem for semistable elliptic curves. Together with earlier work by Ken Ribet, this proved Fermat's Last Theorem. Wiles's proof was published in 1995 and has since influenced many areas of mathematics.

The full modularity theorem was later proved by other mathematicians building on 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 generalizations.

One generalization 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 related 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 explore the limits of 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 a large amount of money—100,000 gold marks—for anyone who could solve it. Many people tried to solve it, but most of their proofs were wrong.

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

In popular culture

Fermat's Last Theorem has captured the imagination of people beyond the world of mathematics, earning a special place 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 incorrect.