# Monthly Archives: August 2013

## A short proof that 2^{\sqrt{2}} is not rational.

Here is an adaptation of Lang’s proof of Gelfond’s theorem to prove that is not rational – needed for http://mathoverflow.net/questions/138247/prove-that-sqrt2-sqrt2-is-an-irrational-number-without-using-a-theorem/138291#138291. Consider the functions . Let , where is big enough (we shall see how large should be below). Suppose that is rational. Then is a rational number for every . Lemma 1. Let us have […]