Simple Tests for n-th Roots of Natural Numbers being Natural Numbers and Elementary Methods to Determine Their Values

Simple Tests for n-th Roots of Natural Numbers being Natural Numbers and Elementary Methods to Determine Their Values Bernd E. Wolfinger

In this paper we tackle the challenging problem to determine, in a simple but reliable way, whether – for a given, arbitrary number x, x \(\geq\) 2 – the n-th root of x produces a rational or an irrational result, i.e. we determine whether \(\sqrt[n]{x}\) \(\epsilon\) Q or \(\sqrt[n]{x}\) \(\notin\) Q. To solve this problem in a straightforward manner we make use of the prime factorization of x. As a main contribution we present a generally applicable algorithm to decide whether \(\sqrt[n]{x}\) \(\epsilon\) Q (for n,x\(\epsilon\)N\{1} ) and if so, to determine the resulting value. Moreover, we design several tests which can be applied to determine, for which values of n, \(\sqrt[n]{x}\) \(\epsilon\) Q if the natural number x satisfies a given set of properties. Quite often the tests proposed will allow us to answer the question “ \(\sqrt[n]{x}\) \(\epsilon\) Q ?” in a matter of seconds. Finally, we demonstrate that, for a very high percentage of all natural numbers x, x \(\geq\) 2, it is impossible to find even a single n \(\epsilon\) N, n \(\geq\) 2 such that \(\sqrt[n]{x}\) \(\epsilon\) Q.
01 27 2024 29 35 10.9734/jamcs/2024/v39i11860 https://journaljamcs.com/index.php/JAMCS/article/view/1860 https://journaljamcs.com/index.php/JAMCS/article/download/1860/3722 https://journaljamcs.com/index.php/JAMCS/article/download/1860/3722 https://journaljamcs.com/index.php/JAMCS/article/download/1860/3723