Quick Overview: Fermat’s Last Theorem

Pierre Fermat was a lawyer and amateur mathematician who died in 1655. He never published his work on mathematics during his life; however, after his death his son published a version of Diophantus’s Arithmetica with his father’s marginal notes. One of these notes read: “Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet” or, in English, “It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.”

In mathematical terms Fermat’s Last Theorem states:

xn+yn=zn  has no non-zero integer solution when n is greater than 2. However, Fermat could not write out his “truly marvelous” proof of this, because the margin was too narrow. Thus, Fermat’s theorem went unpublished and tantalized mathematicians for over 300 years. Until in 1993 Andrew Wiles published a proof of Fermat’s Last Theorem. However, an error was found that may have derailed his proof entirely, but Wiles persisted to correct the mistake. In 1995 Sir Andrew Wiles published his corrected proof of Fermat’s Last Theorem.

Wiles’ full proof (109 pages) can be read at https://www.math.ias.edu/~anindya/fermat.pdf