New Algebraic Proofs on the Correctness of Collatz 
Conjecture basing on New Unified Formulas along with 
Computational Results which Provide Insights on Prime 
Numbers

Yassine Larbaoui

doi:https://doi.org/10.14445/22315373/IJMTT-V72I1P105

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 72 | Issue 1 | Year 2026 | Article Id. IJMTT-V72I1P105 | DOI : https://doi.org/10.14445/22315373/IJMTT-V72I1P105

New Algebraic Proofs on the Correctness of Collatz Conjecture basing on New Unified Formulas along with Computational Results which Provide Insights on Prime Numbers

Yassine Larbaoui

Received	Revised	Accepted	Published
19 Nov 2025	28 Dec 2025	14 Jan 2026	29 Jan 2026

Citation :

Yassine Larbaoui, "New Algebraic Proofs on the Correctness of Collatz Conjecture basing on New Unified Formulas along with Computational Results which Provide Insights on Prime Numbers," International Journal of Mathematics Trends and Technology (IJMTT), vol. 72, no. 1, pp. 28-89, 2026. Crossref, https://doi.org/10.14445/22315373/IJMTT-V72I1P105

Abstract

This paper presents new mathematical proofs on the correctness of the Collatz conjecture based on new unified formulas re-expressing this conjecture in a scalable algebraic form. We are proposing new precise formulas that enable expressing all natural numbers in the Collatz conjecture according to a unified methodology that allows scaling calculations on infinite numbers, and then we use these unified formulas to prove that all natural numbers are converging to the number one when we perform Collatz operations on them. In addition, we are presenting computational codes based on these algebraic formulas to provide computational results that guide the development of proofs. As a result, this paper is presenting twenty seven new theorems along detailed proofs, where the first seven theorems are proposing new unified formulas to re-express the values of numbers according to unified algebraic forms, whereas the other theorems are using these new formulas to demonstrate that all natural numbers are converging to the number “1” when we forward Collatz operations on them; whereas demonstrating that there is no divergency over these operations. We also use these formulas to demonstrate that all natural numbers create loops in infinity that may cause Collatz calculations to circulate in a ring of numbers. Furthermore, we demonstrate that the only ring of natural numbers that creates a loop for the Collatz conjecture is where we go from “1” to “4”, then calculations converge back to the number one. All theorems presented in this paper are developed according to an engineering methodology based on structuring unified formulas to re-express Collatz calculations along step-by-step proofs, which allow us to develop a breakthrough demonstrating the correctness of the Collatz conjecture, while providing new insights into the characteristics of prime numbers and their distribution.

Keywords

Collatz conjecture, New algebraic proofs, New theorems, New unified formulas, Computational codes, Prime numbers characteristics.

References

[1] Jeffrey C. Lagarias, “The 3x+1 Problem and Its Generalizations,” The American Mathematical Monthly, vol. 92, no. 1, pp. 3-23, 1985.
[CrossRef] [Google Scholar] [Publisher Link]

[2] Jean-Paul Delahaye, “The Tenacious Syracuse Conjecture,” For Science, vol. 529, pp. 80-85, 2021.
[Publisher Link]

[3] John H. Conway, “On Unsettleable Arithmetical Problems,” The American Mathematical Monthly, vol. 120, pp. 192-198, 2013.
[CrossRef] [Google Scholar] [Publisher Link]

[4] S. Letherman, and Schleicher Wood, “The (3n + 1)-Problem and Holomorphic Dynamics,” Experimental Mathematics, vol. 8, no. 3, pp. 241-251, 1999.
[CrossRef] [Google Scholar] [Publisher Link]

[5] Jeffrey C. Lagarias, “The 3 X + 1 Problem: An Overview,” The Ultimate Challenge: The 3 x + 1 Problem, Providence, RI, American Mathematical Society, pp. 3-29, 2010.
[Google Scholar] [Publisher Link]

[6] M. Friedewald, “The First Computers-History and Architectures,” IEEE Annals of the History of Computing, vol. 23, no.2, pp. 75-76, 2001.
[CrossRef] [Google Scholar] [Publisher Link]

[7] D.R. Hartree, “The ENIAC, An Electronic Calculating Machine,” Nature, vol. 157, 1946.
[CrossRef] [Google Scholar] [Publisher Link]

[8] Godfrey Harold Hardy, and Edward Maitland Wright, An Introduction to the Theory of Numbers, Oxford University Press, 1979.
[Google Scholar] [Publisher Link]

[9] Shalom Eliahou, “The 3x+1 Problem: New Lower Bounds on Nontrivial Cycle Lengths,” Discrete Mathematics, vol. 118, no. 1-3, pp. 45-56, 1993.
[CrossRef] [Google Scholar] [Publisher Link]

[10] D. Barina, “Improved Verification Limit for the Convergence of the Collatz Conjecture,” The Journal of Supercomputing, vol. 81, pp. 1-14, 2025.
[CrossRef] [Google Scholar] [Publisher Link]

[11] R.E. Crandall, “On the 3x+1 Problem,” Mathematics of Computation, vol. 32, pp. 1281-1292, 1978.
[Google Scholar] [Publisher Link]

[12] David Barina, “Convergence Verification of the Collatz Problem,” The Journal of Supercomputing, vol. 77, no.3, pp. 2681–2688, 2020.
[CrossRef] [Google Scholar] [Publisher Link]

[13] Terence Tao, “Almost All Orbits of the Collatz Map Attain Almost Bounded Values,” Forum of Mathematics, vol. 10, 2022.
[CrossRef] [Google Scholar] [Publisher Link]

[14] Z.J. Hu, “The Analysis of Convergence for the 3X + 1 Problem and Crandall Conjecture for the X + 1 Problem,” Advances in Pure Mathematics, vol. 11, pp. 400-407, 2021.
[Google Scholar]

[15] Oliveira, T. e Silva, “Empirical Verification of the 3x+1 and Related Conjectures. The Ultimate Challenge: the 3x+1 Problem,” American Mathematical Society, Providence, pp.189-207, 2010.
[Google Scholar]

[16] Michael I. Rosen, “Niels Hendrikabel and Equations of the Fifth Degree,” American Mathematical Monthly, vol. 102, no. 6, pp. 495-505, 1995.
[CrossRef] [Google Scholar] [Publisher Link]

[17] Yassine Larbaoui, “New Theorems and Formulas to Solve Fourth Degree Polynomial Equation in General Forms by Calculating the Four Roots Nearly Simultaneously,” American Journal of Applied Mathematics, vol. 11, no. 6, pp. 95-105, 2023.
[CrossRef] [Google Scholar] [Publisher Link]

[18] Yassine Larbaoui, “New Theorems Solving Fifth Degree Polynomial Equation in Complete Forms by Proposing New Five Roots Composed of Radical Expressions,” American Journal of Applied Mathematics, vol. 12, no. 1, pp. 9-23, 2024. 123456.
[CrossRef] [Google Scholar] [Publisher Link]

[19]Yassine Larbaoui, “New Six Formulas of Radical Roots Developed by Using an Engineering Methodology to Solve Sixth Degree Polynomial Equation in General Forms by Calculating All Solutions Nearly in Parallel,” American Journal of Applied Mathematics, vol. 13, no. 1, pp. 73-94, 2024.
[CrossRef] [Google Scholar] [Publisher Link]

[20] Godfrey Harold Hardy, and Edward Maitland Wright, An Introduction to the Theory of Numbers, Oxford University Press, 1979.

[21] Pedro Berrizbeitia, and Boris Iskra, “Gaussian Mersenne and Eisenstein Mersenne Primes,” Mathematics of Computation, vol. 79, no. 271, pp. 1779-1791, 2010.
[Google Scholar] [Publisher Link]

[22] Albert Edward Ingham, The Distribution of Prime Numbers, Cambridge University Press. pp. 2-5, 1990.
[Google Scholar] [Publisher Link]

[23] K. Soundararajan, “The Distribution of Prime Numbers,” Equidistribution in Number Theory, An Introduction, vol. 237, 2007.
[Google Scholar] [Publisher Link]

[24] Abdalbasit Mohammed Qadir, and Nurhayat Varol, “A Review Paper on Cryptography,” 7^th International Symposium on Digital Forensics and Security, Barcelos, Portugal, pp. 1-6, 2019.
[CrossRef] [Google Scholar] [Publisher Link]

[25] David M. Burton, The History of Mathematics: An Introduction, 7^th ed., McGraw-Hill, 2011.
[Google Scholar] [Publisher Link]

[26] Calvin T. Long, Elementary Introduction to Number Theory, 2^nd ed., Lexington: D. C. Heath and Company, 1972.
[Publisher Link]

[27] Mircea Ghidarcea, and Decebal Popescu, “Prime Number Sieving—A Systematic Review with Performance Analysis,” Algorithms, vol. 17, no. 4, pp. 1-20, 2024.
[CrossRef] [Google Scholar] [Publisher Link]

[28] Jerome A. Solinas, “Generalized Mersenne Prime,” Encyclopedia of Cryptography and Security, pp. 509-510, 2011.
[CrossRef] [Google Scholar] [Publisher Link]

[29] Michael McCool, James Reinders, and Arch Robison, Structured Parallel Programming: Patterns for Efficient Computation, 1^st ed., Morgan Kaufmann Publishers Inc., 2012.
[Google Scholar] [Publisher Link]

[30] Chiara Bellotti, “Explicit Bounds for the Riemann Zeta Function and a New Zero-Free Region,” Journal of Mathematical Analysis and Applications, vol. 536, no. 2, pp. 1-33, 2024.
[CrossRef] [Google Scholar] [Publisher Link]

[31] Dave Platt, and Tim Trudgian, “The Riemann Hypothesis is True up to 3.10¹²",” Bulletin of the London Mathematical Society, vol. 53, no. 3, pp. 792-797, 2021.
[CrossRef] [Publisher Link]

[32] S. Dutta, “Chronological Verification of the Collatz Conjecture using Theoretically Proven Sieves,” Electronic Journal of Mathematical Analysis and Applications, vol. 13, no. 1, pp. 1-10, 2025.
[Google Scholar]