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Mathematics of Computation

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

 
 

 

A deterministic algorithm for solving $n=fu^ 2+gv^ 2$ in coprime integers $u$ and $v$


Authors: Kenneth Hardy, Joseph B. Muskat and Kenneth S. Williams
Journal: Math. Comp. 55 (1990), 327-343
MSC: Primary 11Y50; Secondary 11D09
DOI: https://doi.org/10.1090/S0025-5718-1990-1023762-3
MathSciNet review: 1023762
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Abstract: We give a deterministic algorithm for finding all primitive representations of a natural number n in the form $f{u^2} + g{v^2}$, where f and g are given positive coprime integers, and $n \geq f + g + 1$, $(n,fg) = 1$. The running time of this algorithm is at most \[ \mathcal {O}({n^{1/4}}{(\log n)^3}(\log \log n)(\log \log \log n)),\] uniformly in f and g.


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Article copyright: © Copyright 1990 American Mathematical Society