Divisors in residue classes, constructively

Coppersmith, Don; Howgrave-Graham, Nick; Nagaraj, S.

doi:10.1090/S0025-5718-07-02007-8

Divisors in residue classes, constructively
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by Don Coppersmith, Nick Howgrave-Graham and S. V. Nagaraj PDF

Math. Comp. 77 (2008), 531-545 Request permission

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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