Wednesday, November 2, 2011

Fermat's last theorem and the simpsons

Jack Dikian
November 2011

The theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. Fermat's last theorem has been one of the most difficult problems to solve. It took over 350 years to solve and has precipitated more incorrect proofs than any other math problem.

Fermat claimed to have proved this statement but that the margin was too narrow to contain it. It is the seeming simplicity of the problem, coupled with Fermat's claim to have proved it, which has captured the hearts of so many mathematicians.
In 1995 Andrew Wiles released his 108-page proof after spending almost 7 years developing it. The proof is regarded to be one of the most complicated in mathematics and involved (chapters in Wiles’ paper) Galois representations, cohomology groups, Gorenstein property, congruences between Hecke rings, Selmer group, elliptic curves, Gorenstein rings and local complete intersections.

Fermat's last theorem even entered the fictional world of The Simpsons. In the episode “Treehouse of Horror VI” a sum, proved impossible by the theorem, the equation ‪178212 + 184112 = 192212 is visible. The joke being that the twelfth root of the sum does evaluate to 1922 due to rounding errors. Note that the left hand side is odd, while ‪192212 is even, so the equality cannot hold. Instead of 1922, it actually is 1921.999999995.