What We Learn Along the Way: Fermat’s Last Theorem

In 1637 Pierre de Fermat wrote in the margin of Diophantus’s Arithmetica the statement that would puzzle some of the world’s greatest mathematicians for over three centuries:

It is impossible to write a cube as a sum of two cubes, a fourth power as a sum of two fourth powers, and, in general, any power beyond the second as a sum of two similar powers. For this, I have discovered a truly wondrous proof, but the margin is too small to contain it.

Fermat, celebrated for making such declarations with little confirmation, kept mathematicians scratching their heads and squaring their roots trying to discover proofs for his statements. By the 19^th century, all of Fermat’s theories had been resolved except the one above, a statement that became know as Fermat’s Last Theorem.

We will never know whether Fermat had actually discovered a correct proof of his theorem, but we do know that Andrew Wiles of Princeton University produced a 130-page proof in 1994, 357 years after Fermat wrote his tantalizing marginal note.

Although Fermat’s Last Theorem has not yet been used for practical purposes, many new ideas and numerous practical technological advances developed in solving the problem. Sometimes what we learn along the way to a destination becomes more important than reaching the end of our journey.  

Revised from The Heart of Mathematics by Edward Burger and Michael Starbird