Pierre de Fermat : Three Century Old Puzzle

Pierre de Fermat is a French mathematician in the 17th century.

In 1637, While studying book II of the Arithmetica he came upon a whole series of observations, problems, and solutions that concerned Pythagoras’s theorem and Pythagorean triples.

Instead of considering the equation:


He contemplated a variant of the Pythagoras’s creation:


Just by changing the exponent from 2 to 3 and he already had the impression that this equation had no integer solution whatsoever. He wondered if could it really be the case that this minor modification turns Pythagoras’s equation, one with an infinite number of solutions into an equation with no integer solutions?

Fermat altered the equation further by changing the exponent to numbers bigger than 3, and thought that the resulting equations had no integer solutions. In other words, the following equation:


where n>2

had no integer solutions.

In the margin of his Arithmetica, he wrote down the following note:

Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere

which translates into:

It is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as the sum of two fourth power or, in general, for any number which is a power greater than the second to be written as the sum of two like powers.

It was an extraordinary claim that Fermat believed he could prove. After the note outlying the theory, the mischievous genius wrote down the additional comment below:

Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet

which translates into:

I have a truly marvelous demonstration of this proposition which this margin is too small to contain it.

This small note became known as Fermat’s Last Theorem

There is no integer solution to the equation xn+yn=zn

where n>2

And there it is ladies and gentlemen, unbeknownst to Fermat, he had created the world’s hardest mathematical problem, a deceptively simple one that defeated the greatest minds in mathematics for over 300 years!! because it wasn’t until 1994, a good 357 years since its creation by Fermat, that my intellectual hero the brilliant British mathematician Sir Andrew John Wiles created a highly sophisticated proof spanning 200 pages that solved this famous puzzle. With this proof, it is now a fact that there is no integer solution to the Fermat’s equation.

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