So I recently came across a very ridiculous proof on Quora that I thought was very cool. This proof uses the infamous Fermat’s Last Theorem to create a very slick proof of a typical problem. I would have never fathomed that such a behemoth of a theorem could be used to solve a very simple-looking problem. Here it is below:
Prove that the cube root of 2 is irrational.
So let’s suppose by proof of contradiction that the cube root of 2 is in fact rational.
Therefore 2^(1/3) = p/q, where both p and q are rational numbers.
By cubing both sides, we get that 2 = p^3 / q^3, or p^3 = 2q^3.
We can modify the equation so that it becomes p^3 = q^3 + q^3. We know that this equation is false because of Fermat’s Last Theorem, in which a specific case of the theorem states that a^3 + b^3 ≠ c^3. Therefore, the cube root of 2 must be irrational!
To generalize, this proof also works for any nth root of 2, where n is an integer greater than or equal to 3.
Like I said, I really enjoyed reading this proof because it used something so esoteric to solve a very simple problem. I guess this goes to show that everything has a purpose.