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.

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February 18, 2008 at 12:35 am

John smithso we are turning a simple problem into a much harder one (flt)?

At any rate, how many of us could actually prove flt?

February 18, 2008 at 1:46 am

VishalWell, you are right of course! However, I just thought now that we do know FLT

istrue, it is kind of cool to prove the irrationality of the root of using FLT!February 18, 2008 at 3:03 am

John ArmstrongWow. And Michael tears into

mefor using (any form of) the Mean Value Theorem in proving the Fundamental Theorem of Calculus.February 18, 2008 at 3:04 am

VishalI know! 😀