TV Documentary on Fermat’s Last Theorem

This is the first part of a TV documentary titled Fermat’s Last Theorem
made by the famous British science author, Simon Singh, along with John Lynch in 1996. More information on the documentary can be found here.

To quote Simon Singh,

In 1996, working with John Lynch, I made a documentary about Fermat’s Last Theorem for the BBC series Horizon. It was 50 minutes of mathematicians talking about mathematics, which is not the obvious recipe for a TV blockbuster, but the result was a programme that captured the public imagination and which received critical acclaim.

The programme won the BAFTA for best documentary, a Priz Italia, other international prizes and an Emmy nomination – this proves that mathematics can be as emotional and as gripping as any other subject on the planet…

You should definitely watch the opening scene which became very famous later on!
Part 1:

There are five parts in all. The rest of the four may be found by visiting YouTube.

Irrationality of nth root of 2

The following fun problem was posed in one of the issues of the American Mathematical Monthly (if I am not wrong). I don’t remember the exact issue or the author, but here is the problem anyway.

Prove that is irrational for all and .

Slick solution: We could either use Euclid’s arguments or invoke the rational root theorem to prove the above statement. However, there is a slicker proof!

Assume, for the sake of contradiction, that , where and . Then, we have which implies . But this contradicts Fermat’s Last Theorem! And, we are done.