Fermat’s Last Theorem

All of us are aware of Pythagoras’s Theorem which says that if a, b & c are 3 sides of a right angle, then a2 + b2 = c2. Here a, b, & c have to be whole numbers. There are many sets of triples (a, b & c) which obey such a relation. The set of numbers which obey this relation are called Pythagorean Triples. Some examples are 3, 4 & 5 and 5, 12 & 13.

32=9, 42 = 16, 52 = 25 and 9 + 16 = 25

52 = 25, 122 = 144, 132 = 169 and 25 + 144 = 169

French mathematician, Pierre Fermat was interested to find out if there were powers more than 2 for which such a relation would be true. i.e would there be 3 numbers which would satisfy a3 + b3 = c3? Or for any higher power? For a long time, Fermat could not find triples which satisfied this relation for powers more than 2. But not finding such triples is no proof that they do not exist. We have seen that mathematicians need a rigorous proof that such triples do not exist.

In 1650 Fermat made a cryptic noting in the margin of a book that he was reading. The noting said that he had found a proof that such a relation would not be possible for powers more than 2. He also added that the margin was too small to write the details!

This made the problem famous and many famous mathematicians started working on it. Many of them proved specific powers for which they could not exist. For example, one mathematician proved that such triples did not exist for all powers less than 100, except for 37, 59 & 67.

Sophie Germain, the female French mathematician, was the first person to make substantial contributions to the eventual solution of Fermat’s Last Theorem, introducing several new ways to rephrase the theorem and a partial solution dealing with integers less than 100. This fact was found out, after her death, through her diary foot notes!

The proof took more than 350 years to be finalised! In 1994, Andrew Wiles became world famous for providing a proof which was studied and accepted by mathematicians.

The proof brought him the Abel prize in 2016 and other awards.

More surprising was the fact that the proof used many new areas of math which were invented after the problem was formulated and which did not seem to have any connection with the simple statement of the problem. Wile’s proof used so many new areas of math and was so long, that only a few mathematicians had the knowledge to go through the proof, understand it and declare that it was correct.

We can see that the problem is simple enough to be understood by a Class 5 student. But it took the effort of many mathematicians almost 350 years to arrive at a proof.

Finally, an explanation of the term “Last” in the title of this theorem. It was not the last theorem written by Fermat. Fermat proposed and proved many theorems in a branch of math called Number Theory. All his theorems and proposals were proved. The “last” theorem was the last of his theorem which was not proved for almost 350 years!