When we think of proofs, we think of long, esoteric manipulations, yet sometimes they have quite a surprising simplicity (which is one of the reasons they are so intriguing). Here’s a conjecture which has a really cool proof to it.
There exist irrational numbers x and y such that xy gives a rational number.
Let’s start with the most well-known irrational number: √2. What is (√2)√2? If it is rational, then the conjecture is proved, but this is not a trivial thing to find out. So let’s assume it’s irrational. What happens when we raise it to the power √2? That’s ((√2)√2))√2, which results in (√2)2, which is 2, a rational number. And that’s it. If (√2)√2 is rational, it is proved, and if it is irrational, then we can raise it to the power root two, which is an irrational number to the power an irrational number, and we get a rational number. Thus the theorem is proved. I know, this was pretty awesome.