Sufficient Conditions for n-th Root of x Being an Irrational Number for all n\(\in\)N, n\(\ge\)2
Bernd E. Wolfinger *
Computer Science Department, University of Hamburg, Germany.
*Author to whom correspondence should be addressed.
Abstract
The challenging question “Given a number x\(\in\)N, x\(\ge\)2: For which n\(\in\)N, n\(\ge\)2 the value of \(\sqrt[n]{x}\) is a rational number?” has been solved completely if the prime factorization of x is available. However, prime factorization may be impossible in practice, in particular for extremely large numbers. Therefore, in this paper we present various sufficient conditions which allow us to prove that “\(\sqrt[n]{x}\) \(\notin\) Q, \(\forall\)n \(\in\) N, n\(\ge\)2” just based on the knowledge of only one or two factors of the prime factorization of x.
Keywords: Number theory, n-th roots of real numbers, irrational n-th roots, sufficient conditions for irrational roots