The World’s Most Difficult Problem- The story of Fermat and his last theorem.

Fermat’s Last Theorem is a theorem that even a high-school kid can understand, but whose solution had eluded the mathematical community for nearly 385 years. The theorem was even featured in the Guinness Book of World Records as the world’s “ most difficult mathematical problem ”.

Pierre de Fermat, when not arguing cases in the French judicial courts was mostly found pursuing his copy of Arithmetica, a Greek mathematics book detailing various algebraic and geometric problems. It was at a time when the pursuit of mathematics as a academic field was not really encouraged. Additionally, French mathematicians were known to be extremely secretive regarding their solutions- mainly because at that point in time, their promotions and demotions hinged on their ability to defeat their rivals in a mathematical duel. Fermat, on the other hand, was different. He preferred to keep out of this rough competition and was content with devising theorems to prove abstract concepts. He would even write letters to other mathematicians sometimes- detailing the problem he was facing and challenging them to find a solution. Needless to say, these overtures were not fondly remembered by his colleague.

After his death Fermat’s eldest son decided to go through his father’s inheritance. While doing so he discovered that on the margin of Fermat’s copy of the Arithmetica, there were lots of scribbles detailing the various abstract problems that Fermat had come across. Fermat’s eldest son then decided to publish Fermat’s problems in a special version of the Arithmetica. Once Fermat’s problems reached the wider mathematical community, the mathematicians set about filling in the gaps left by Fermat. Fermat however, hadn’t provided the proofs and solutions to any of his problem, rather he had left tantalizing hints which were enough to convince others that he had the solution.

Almost all of Fermat’s problems were solved- except one. The problem was jotted down beside Diophantus’s explanation of the Pythagorean triplets. There was also a note, and it said :

” 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.”

Despite the rather confusing language, the theorem is rather simple. It states that there are no solutions for the equation

a^n + b^n = c^n when n > 2.

This was Fermat’s last theorem. As seen in the note, he provided a tantalizing hint that he had the solution, yet he choose not to jot it down as “this margin is too narrow to contain [it] ”. Despite the efforts of many well known mathematicians, this theorem remained unsolved for 385 years. Paul Friedrich Wolfskehl, a German industrialist and amateur mathematician bequeathed a prize of 1000,000 marks (a lot of money in those days) to the person who would solve the theorem. There is a lot of speculation as to how Wolfskehl came across the theorem, the most popular of which is that the theorem distracted him from suicide, something he had contemplated after being rejected by a young lady. Hence, he wished to show his gratitude in the form of the prize. 

 

Mathematics professor Andrew Wiles is seen here with the theorem, which he solved, written on a chalkboard in his Princeton, N.J., office, back in 1998.

Finally, in 1917 a mathematician named Andrew Wiles solved the theorem using techniques that were not known during the time of Fermat. It was a extremely sad day for the mathematical community when they learnt that the creator of the “ worlds toughest problem” didn’t actually have the correct proof in reality.

 

This article was contributed by Edudigm’s student Sohom Datta

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