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Mathematics of Computation

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Speeding Fermat's factoring method

Author: James McKee
Journal: Math. Comp. 68 (1999), 1729-1737
MSC (1991): Primary 11Y05; Secondary 11Y16, 68Q25
Published electronically: March 1, 1999
MathSciNet review: 1653962
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Abstract: A factoring method is presented which, heuristically, splits composite $n$ in $O(n^{1/4+\epsilon})$ steps. There are two ideas: an integer approximation to $\surd(q/p)$ provides an $O(n^{1/2+\epsilon})$ algorithm in which $n$ is represented as the difference of two rational squares; observing that if a prime $m$ divides a square, then $m^2$ divides that square, a heuristic speed-up to $O(n^{1/4+\epsilon})$ steps is achieved. The method is well-suited for use with small computers: the storage required is negligible, and one never needs to work with numbers larger than $n$ itself.

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Additional Information

James McKee
Affiliation: Pembroke College, Oxford, OX1 1DW, UK

Received by editor(s): March 28, 1997
Published electronically: March 1, 1999
Article copyright: © Copyright 1999 American Mathematical Society