Canonical form of modular hyperbolas with an application to integer factorization

01/23/2020

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by   Juan Di Mauro, et al.

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For a composite n and an odd c with c not dividing n, the number of solutions to the equation n+a≡ b c with a,b quadratic residues modulus c is calculated. We establish a direct relation with those modular solutions and the distances between points of a modular hyperbola. Furthermore, for certain composite moduli c, an asymptotic formula for quotients between the number of solutions and c is provided. Finally, an algorithm for integer factorization using such solutions is presented.