Authors: Richard Beigel
Abstract: Let k and t be positive integers. It is well known that k1/t is either a positive integer or an irrational number. This can be proved easily from the unique factorization theorem.
An interesting question is how much number theory is necessary in order to prove this theorem. Maier and Niven and Floyd have presented proofs of this theorem that use no facts about prime numbers. The former use the division algorithm and induction to simplify and generalize Steinhaus's proof of the case k = t = 2; the latter uses the Euclidean algorithm to simplify and generalize Sagher's proof of the case t = 2. When t = 2, Maier and Niven's technique is especially interesting because it does not explicitly use any number theory, only very basic inequalities.
In this paper, we also prove the theorem for all k and t. Although our proof is more complicated than Floyd's, it has the advantage of not explicitly using any number theory, so it can be presented to a very general audience. Our proof is simpler than Maier and Niven's.