Fermat's Last Theorem

The idea

The equation xⁿ + yⁿ = zⁿ has no solution in natural numbers x, y, z when n > 2: no cube is a sum of two cubes, no fourth power a sum of two fourth powers, and so on.

From squares to cubes

Squares can add up to a square: 3² + 4² = 5². There are infinitely many such triples.

What about cubes? Is there a cube that is a sum of two cubes: x³ + y³ = z³, with whole numbers x, y, z?

What do you think?

• Yes, many, just like squares
• Yes, but only a few
• No, none at all

Squares, cubes, fourth powers

Choose a power n and two numbers x and y. The toy adds xⁿ + yⁿ and checks if the total is an exact n-th power. Start with n = 2, then try n = 3 and n = 4.

Fermat's Last Theorem

The equation xⁿ + yⁿ = zⁿ has no solution in natural numbers x, y, z when n is bigger than 2. No cube is a sum of two cubes. No fourth power is a sum of two fourth powers. And so on.

In the 17th century Fermat wrote in a book's margin that he had "a truly marvellous proof", but the margin was too small. Nobody found it. After more than 300 years of failed attempts, Andrew Wiles proved it in 1994. He first read about the problem as a 10-year-old.

A search of small numbers is not a proof. 6³ + 8³ = 728 misses 9³ = 729 by only 1, and 9³ + 10³ = 1729 misses 12³ = 1728 by 1. Only a proof covers every x, y, z.

Notes

Fermat's Last Theorem. The equation xⁿ + yⁿ = zⁿ has no solution in natural numbers when n > 2. Squares work (n = 2); cubes, fourth powers and higher never do. Proved by Andrew Wiles in 1994.

Check yourself

Which of these equations has solutions in natural numbers?

What is 6³ + 8³? (6³ = 6 × 6 × 6 and 8³ = 8 × 8 × 8.)

Answer: 728

216 + 512 = 728. This is not a cube: 8³ = 512 is too small and 9³ = 729 is just one more.

9³ = 729. By how much does 6³ + 8³ = 728 miss it?

Answer: 1

729 − 728 = 1. So close, but it is not a solution.

We tried many pairs with n = 3 and found no solution. Does that prove there is none?

• x³ + y³ = z³. Fermat's Last Theorem says there is no solution when n is 3 or more.
• x² + y² = z² — correct. Yes! Baudhāyana triples such as (3, 4, 5) solve it, and there are infinitely many.
• x⁴ + y⁴ = z⁴. Fermat's Last Theorem says there is no solution when n is 3 or more, and 4 is more than 2.
• No. There are infinitely many pairs, and only a proof covers all of them. Wiles gave one in 1994. — correct. Yes! Testing can find a solution, but it can never check every x, y, z. That needs a proof.
• Yes, because we tried enough pairs.. However many we try, there are always more. Only a proof covers all the numbers.
• Yes, because Fermat said so.. Fermat said he had a proof, but nobody ever found it. The theorem was proved by Wiles in 1994.