Factorization of Integers

Frantz Olivier

doi:https://doi.org/10.14445/22315373/IJMTT-V72I3P104

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 72 | Issue 3 | Year 2026 | Article Id. IJMTT-V72I3P104 | DOI : https://doi.org/10.14445/22315373/IJMTT-V72I3P104

Factorization of Integers

Frantz Olivier

Received	Revised	Accepted	Published
17 Jan 2026	22 Feb 2026	13 Mar 2026	27 Mar 2026

Citation :

Frantz Olivier, "Factorization of Integers," International Journal of Mathematics Trends and Technology (IJMTT), vol. 72, no. 3, pp. 23-39, 2026. Crossref, https://doi.org/10.14445/22315373/IJMTT-V72I3P104

Abstract

Considering the breakthrough that happened recently in math, which established the condition for primality versus composite numbers, the decomposition of large numbers has been put to rest. To fully comprehend factorization of integers, this paper presents this issue with regard to the area of cryptography. A cryptosystem has two primary functions: to encipher, which means to prepare the message in a coding scheme, and to decipher the information, which means to decode the message. This decryption cannot be accomplished without knowledge of the secret deciphering key. For this reason, it is natural to look at an ancient problem of number theory, the problem of finding the complete factorization of a large composite integer whose prime factors are not known in advance. To reach this goal, an algebraic system is presented herein that is robust enough to handle large numbers. This system will encompass all that we know about algebraic properties, including the integration of some known theories of integers (e.g., lattice points representation). Parametric representation of curves as single points is necessary because these curves, after analysis, will be a piece of information toward the limit of the function f. A pattern will be established using the gap between a point on the rational plane and a point on the irrational plane. In doing so, it will be proven that the curves under study converge to the curve that holds the key to factorization. A demo is available that will perform the principle of factorization in nlogn complexity (i.e., the length of the input of integers versus the running time of the output of the function). This is called the time complexity of the algorithm, and it will be shown, herein, that indeed it is nlogn complexity. This work is designed to answer related problems in the field of mathematics, such as trisecting angles in geometric figures, banking issues such as transfer of funds, and communication integrity. However, the strongest modern algorithms (quadratic sieve, elliptic-curve method, and number field sieve) have been unable to resolve any of these ten numbers. It can be stated that the current effective limit of systematic factoring is ~100 digits, but it is still instructive and rewarding to find factors of much larger numbers. This is where a sandwich could make a difference; where a triangulation can take place to control false readings in conjunction with backtracking, leveraging new computers and graphic abilities. This new math breakthrough that has occurred enables the capability of distinguishing prime numbers from composite numbers. Resolving the latter into its prime factors is the last hurdle that needs to be crossed.

Keywords

Integers, RHO, Gauge Point Calibration.

References

[1] Alfred V. Aho et al., Data Structures and Algorithm, Pearson; 1^st Edition, 1983.
[Google Scholar] [Publisher Link]

[2] John C. Amazigo, Advanced Calculus and Its Applications to the Engineering and Physical Sciences, John Wiley and Sons Publications, 1980.
[Publisher Link]

[3] Frank Ayres, and Elliott Mendelson, Schaum’s Outline of Calculus, McGraw-Hill, pp. 1-578, 2000.
[Google Scholar] [Publisher Link]

[4] Thomas Becker, and Volker Weispfenning, Grobner Bases A Computational Approach to Commutative Algebra, Springer, Corrected Edition, 1993.
[Publisher Link]

[5] Harold P. Boas, “A Worm?,” Notices of the American Mathematical Society, vol. 50, no. 5, pp. 554-555, 2003.
[Google Scholar]

[6] Folkmar Bornemann, “PRIMES Is in P: Breakthrough for “Everyman,” Notices of the American Mathematical Society, vol. 50, no. 5, pp. 545-553, 2003.
[Google Scholar] [Publisher Link]

[7] Samuel R. Buss, “Bounded Arithmetic,” University of California, Berkley, 1985.
[Google Scholar] [Publisher Link]

[8] Harvey Cohn, Advanced Number Theory, Dover Publications, 1980.
[Publisher Link]

[9 Richard E. Crandall, Projects in Scientific Computation, Springer-Verlag New York, 1994.
[Publisher Link]

[10] Nell B. Dale, and David Orshalick, Introduction to PASCAL and Structured Design, D.C. Heath, pp. 1-579, 1983.
[Google Scholar] [Publisher Link]

[11] Reinhard Diestel, Graph Decompositions A Study in Infinite Graph Theory, Oxford University Press, 1990.
[CrossRef] [Google Scholar] [Publisher Link]

[12] A. Evyatar, and P. Rosenbloom, Motivated Mathematics, Cambridge University Press, 1981.
[Google Scholar] [Publisher Link]

[13] John W. Keese, Elementary Abstract Algebra, Heath, pp. 1-168, 1965.
[Google Scholar] [Publisher Link]

[14] Donald E. Knuth, The Art of Computer Programming, vol 1: Fundamental Algorithms, 2^nd Edition, Addison-Wesley, 1973.
[Publisher Link]

[15] Neal Koblitz, Graduate Texts in Mathematics, A Course in Number Theory and Cryptography, Springer, Second Edition, 1994.
[CrossRef] [Google Scholar] [Publisher Link]

[16] Jean Louis Lassez, Gordon Plotkin, and J.A. Robinson, Computational Logic Essays in Honor of Alan Ronbinson, The MIT Press, 1991.
[Google Scholar] [Publisher Link]

[17] Harry P. Lewis, and Christos H. Papadimitrious, “Element of the Theory of Computation,” ACM SIGACT News, vol. 29, no. 3, pp. 62-78, 1998.
[Google Scholar] [Publisher Link]

[18] M.J. Maron R.J. Lopez, Numerical Analysis: A Practical Approach, Harcourt Publishers, 1991.
[Publisher Link]

[19] J. L. Brenner, and Douglas B. West, “The American Mathematic Monthly,” Taylor and Francis, vol. 94, no. 10, pp. 997-1000, 1987.
[Google Scholar] [Publisher Link]

[20] Ivan Niven, Numbers: Rational and Irrational, The L.W. Singer Company, New Mathematical Library, 1961.
[Google Scholar] [Publisher Link]

[21] Piergiorgio Odifreddi, Logic and Computer Science, Academic Press, pp. 1-430, 1990.
[Google Scholar] [Publisher Link]

[22] Vera Pless, Introduction to The Theory of Error-Correcting Codes, Third Edition, Wiley-Interscience Publication, 1998.
[Google Scholar] [Publisher Link]

[23] George Polya, How to Solve It A New Aspect of Mathematical Method, Second Edition, Princeton University Press, 1971.
[Publisher Link]

[24] William F. Trench, Advanced Calculus, Harper and Row, 1978.
[Publisher Link]

[25] Raymond Turner, Truth and Modality for Knowledge Representation, MIT Press, 1990.
[Google Scholar]

[26] Henk C. A. Tilborg, An Introduction to Cryptography, 1^st ed., Springer, pp. 1-170, 1988.
[CrossRef] [Google Scholar] [Publisher Link]

[27] Lincoln A. Wallen, Automated Deduction in Non-Classical Logic, MIT Press, 1989.
[Google Scholar] [Publisher Link]

[28] Martin M. Zuckerman, Sets and Transfinite Numbers, Macmillan, pp. 1-423, 1974.
[Google Scholar] [Publisher Link]