Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# 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

Comment 1
Commenter:
The commenter has declared there is no conflict of interests.
Comment: Great! So the famous Fermat number can be easily factorized in the future.
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Comment 2
Commenter: Joseph.Lee
The commenter has declared there is no conflict of interests.
Comment: The paper describes a new method for factorization of odd Integers. The topic is novel and the application proposed is novel as well. The author has made sufficient works according to the goal.The scientific &experimental set-up is valid and the data is presented clearly, and analyzed in detail. In addition, there is experimental comparison of the algorithm with previously known work, and the experimental results show good and new results, so it is easy to judge the algorithm is a better improvement on previous work. For these reasons, I suggest that this paper would be accepted without further modification.
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Comment 3
Commenter:
The commenter has declared there is no conflict of interests.
Comment: The paper describes a new method for factorization of odd Integers. The topic is novel and the application proposed is novel as well.The author has supplied the mathematical proof for the ideas of new method, In addition, the algorithm is experimentally compared with previously known work, and the experimental results show good and new results, so it is easy to judge that the algorithm is a better improvement on previous work.For these reasons, I suggest that this paper would be accepted.
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