Some computational results on a problem concerning powerful numbers

Abstract

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