pk for every non-square positive integer k. It turns out that Estermann’s idea can be further extended to conclude that n p k is either an integer or irrational, for every pair k, n of positive integers. Even more is true: Estermann’s idea leads to a simple proof of Gauss’s Lemma: THEOREM 1. (GAUSS’S LEMMA) Every real root of a monic polynomial with integer coefficients is either an integer or irrational. For the case of n p k, apply the theorem to the monic polynomial P(x) = x n k.

http://www.math.yorku.ca/Who/Faculty/Brettler/2200_15/Gilat.pdf

Gauss's Lemma and the Irrationality of Roots, Revisited

82.10 Estermann and Pythagoras

83.26 More on Estermann and Pythagoras

We now attempt to tile the diagonal of the large rectangle with copies of the small rectangle. One possibility is that, as in Figure 2(a), we exactly reach the bottom right-hand corner in which case (looking at the shorter sides), divides so that is an integer. Alternatively, as in Figure 2(b), we stop short of the bottom right-hand corner when the shaded rectangle left is strictly smaller than the rectangle, but is similar to it and with integer side-lengths. This is a contradiction – either because the argument may be repeated to give an infinite decreasing sequence of whole numbers or, by starting with , the side-lengths of the smallest rectangle with . b a N = a b

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0025557218000657

T. Estermann

Papers

59.3. The irrationality of √2