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Mathematics of Computation

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Divisors in residue classes, constructively

Authors: Don Coppersmith, Nick Howgrave-Graham and S. V. Nagaraj
Journal: Math. Comp. 77 (2008), 531-545
MSC (2000): Primary 11Y05, 11Y16, 68Q25; Secondary 11Y11, 68W40
Published electronically: May 14, 2007
MathSciNet review: 2353965
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Abstract: Let $ r,s,n$ be integers satisfying $ 0 \leq r < s < n$, $ s \geq n^{\alpha}$, $ \alpha > 1/4$, and let $ \gcd(r,s)=1$. Lenstra showed that the number of integer divisors of $ n$ equivalent to $ r \pmod s$ is upper bounded by $ O((\alpha-1/4)^{-2})$. We re-examine this problem, showing how to explicitly construct all such divisors, and incidentally improve this bound to $ O((\alpha-1/4)^{-3/2})$.

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Additional Information

Don Coppersmith
Affiliation: Institute for Defense Analyses, 805 Bunn Drive, Princeton, New Jersey 08540

Nick Howgrave-Graham
Affiliation: NTRU Cryptosystems, 35 Nagog Park, Acton, Massachusetts 01720

S. V. Nagaraj
Affiliation: 66 Venkatrangam Street, Triplicane, Chennai 600 005, India

Keywords: Divisors, residue classes
Received by editor(s): June 5, 2006
Received by editor(s) in revised form: November 14, 2006
Published electronically: May 14, 2007
Article copyright: © Copyright 2007 American Mathematical Society