Wednesday, August 24, 2016

Fermat's Last Theorem - a Tree house of Horror

The popularity of The Simpsons is well known. It is after all the longest-running sitcom of all time. It’s canny; it’s also one of the most literate television programs on air - containing many references to subject matter and scholars from various academic fields. One of the instances of mathematics appears in the "Treehouse of Horror VI" episode.

The equation 178212 +184112 =192212 is visible, just as the dimension begins to collapse. The joke is that the twelfth root of the sum does evaluate to 1922 due to rounding errors when entered into most handheld calculators; notice that the left hand side is odd, while 192212 is even, so the equality cannot hold. Instead of 1922, it is approximately 1921.99999996.

Some will recognize this as Fermat's Last Theorem. The theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than two. You may also know this is one of the most famous math problems in history, as it remained unsolved for well over 300 years.

In the margin of his copy of a book by Diophantus, Pierre de Fermat wrote that it is possible to have a square be the sum of two squares, but that a cube can not be the sum of two cubes, nor a fourth power be a sum of two fourth powers, and so on.

Given that there are infinitely many possible numbers to check it was quite a claim, but Fermat was absolutely sure that no numbers fitted the equation because he had a logical watertight argument. Sadly, he never wrote down his proof. Instead, in the margin of a book, he left a tantalizing note in Latin: “I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.”

Although this is easily stated, it has proved to be one of the most puzzling problems in the whole history of mathematics. Long after all the other statements made by Fermat had been either proved or disproved, this remained.

The correct proof came in May 1995 by Andrew Wiles. Wiles had stumbled upon the last theorem as a 10-year-old and then spent the next 30 years working on the problem. A childhood dream evolved into an adult obsession, and when he eventually figured out a possible strategy for proving Fermat’s riddle, he worked in secrecy for seven years before revealing his 200-page proof.

The proof ultimately uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques not available to Fermat.

Of course there are still problems out there to solve. In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts established seven Prize Problems. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling. There is a $1 million reward for each of these so-called Millennium Problems.

Of the original seven Millennium Prize Problems set by the Clay Mathematics Institute, six have yet to be solved. These are:

  • P versus NP
  • Hodge conjecture
  • Riemann hypothesis
  • Yang–Mills existence and mass gap
  • Navier–Stokes existence and smoothness
  • Birch and Swinnerton-Dyer conjecture