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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Some computational results on a problem concerning powerful numbers

Authors: A. J. Stephens and H. C. Williams
Journal: Math. Comp. 50 (1988), 619-632
MSC: Primary 11R11; Secondary 11A51, 11R27, 11Y16, 11Y40
MathSciNet review: 929558
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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\vert 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}$.

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Article copyright: © Copyright 1988 American Mathematical Society

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