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 2^{\sqrt{2}} 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 f_1=e^{\sqrt{2}x}, f_2=e^x. Let x_i=i\log 2, 1\le i\le m, where m is big enough (we shall see how large m should be below). Suppose that 2^{\sqrt{2}} is rational. Then f_i(x_j) is a rational number for every i,j.

Lemma 1. Let us have a homogeneous system of r linear equations with n variables with integer coefficients, the abs. value of each coefficient is at most A. Then there exists a non-trivial integer solution with |x_i|\le 2(nA)^{\frac{r}{n-r}} provided n>r.

Proof (half a page) see in Lang “Algebra”.

Consider the function F=\sum_{i,j=1}^r b_{i,j}f_1^if_2^j with rational coefficients. Set n=\frac{r^2}{2m} (choose r so that it is an integer). By Lemma 1, we can find integers b_{i,j} such that all derivatives D^k F(x_i)=0, 0\le k<n, i=1,...,m, moreover absolute values of b_{i,j} are bounded by something like O(n^{2n}) for very large n by Lemma 1.  Since f_1, f_2 are algebraically independent  (here we use  that \sqrt{2} is irrational), the function F is not identically 0. Let s\ge n be the smallest integer such that all derivatives up to the order s-1 of F are 0 at each x_i, but the derivative g=D^s(F(x_i))\ne 0 for some i (for simplicity let i=1, the proof is the same for any i). Then g belongs to the field K=\mathbb{Q}[\sqrt{2}], that is g=a+b\sqrt{2} where a,b are rational numbers. Moreover the least common denominator c of a, b  is bounded by C_1^s for some constant C_1. The conjugate of cg (i.e.,ca-cb\sqrt{2}) is bounded by C_2s^{5s} for some constant C_2. Then the norm of cg in K, i.e. the product of cg and its conjugate,  which is an integer \ge 1, is bounded by C_2s^{5s}|g|. Now consider the function H(z)=\frac {F(z)}{\prod_{i=1}^{m}(z-x_i)^{s-1}}. Then H(x_1) differs from D^sF(x_1) by some factor bounded by C_3^s s! (which is “small”). By the maximum modulus principle, the value of H(x_1) is bounded above by the value of H(z) where z is a complex number on a circle of large enough radius R (we need only that R is bigger than each x_i). Taking R large enough, we get that H(x_1) is bounded from above by a number of the form \frac{s^{3s}C_4^{2rR}}{R^{ms}}. For R=s^{C_4s} (C_4 is an appropriate constant) this gives a contradiction with the previous inequality for |g| (the denominator grows much faster than the numerator, so the fraction cannot be always greater than 1).

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