Some Computational Results on a Problem Concerning Powerful Numbers

Abstract

Let D be a positive square-free integer and let $X + Y\sqrt D$ be the fundamental unit in the order with Z-basis $\{ 1,\sqrt D \}$. An algorithm, which is of time complexity $O({D^{1/4 + \varepsilon }})$ for any positive $\varepsilon$, is developed for determining whether or not $D|Y$. Results are presented for a computer run of this algorithm on all $D < {10^8}$. The conjecture of Ankeny, Artin and Chowla is verified for all primes $\equiv 1 \pmod 4$ less than ${10^9}$.