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Factorization of Odd Integers with a Divisor of the Form $2^au \pm 1$
Version 1
: Received: 10 March 2020 / Approved: 11 March 2020 / Online: 11 March 2020 (02:53:39 CET)
How to cite: Wang, X. Factorization of Odd Integers with a Divisor of the Form $2^au \pm 1$. Preprints 2020, 2020030174. https://doi.org/10.20944/preprints202003.0174.v1 Wang, X. Factorization of Odd Integers with a Divisor of the Form $2^au \pm 1$. Preprints 2020, 2020030174. https://doi.org/10.20944/preprints202003.0174.v1
Abstract
The paper proves that an odd composite integer $N$ can be factorized in at most $O( 0.125u(log_2N)^2)$ searching steps if $N$ has a divisor of the form $2^a{u} +1$ or $2^a{u}-1$ with $a > 1$ being a positive integer and $u > 1$ being an odd integer. Theorems and corollaries are proved with detail mathematical reasoning. Algorithms to factorize the kind of odd composite integers are designed and tested by factoring certain Fermat numbers. The results in the paper are helpful to factorize the related kind of odd integers as well as some big Fermat numbers
Keywords
integer factorization; Fermat number; cryptography; algorithm
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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