Some computational results on a problem concerning powerful numbers

Stephens, A. J.; Williams, H. C.

doi:10.1090/S0025-5718-1988-0929558-3

Some computational results on a problem concerning powerful numbers
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by A. J. Stephens and H. C. Williams PDF

Math. Comp. 50 (1988), 619-632 Request permission

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 ${10^9}$.

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