A new Base-X Conjecture was introduced in this paper, and Collatz Conjecture is just one case of Base-X Conjecture - Base-3 (Ternary). Based on Base-X number system property and Collatz Conjecture iteration, it has been proved that for any positive integer D, there are n and m which exist for
.
is just the result built up by collecting divided by 2 of Collatz Conjecture iteration. Divided by 2m will make the Collatz Conjecture get a result of 1 for any positive integer. Also, the Collatz Tree showed that for any odd positive number, there is only one route existing in the Collatz Tree down to 1 on Collatz Conjecture iteration.
References
[1]
Lagarias, J.C. (2011) The 3x + 1 Problem: An Annotated Bibliography (1963-1999). https://arxiv.org/pdf/math/0309224v13
[2]
Lagarias, J.C. (2012) The 3x + 1 Problem: An Annotated Bibliography, II (2000-2009). https://arxiv.org/pdf/math/0608208
[3]
Guy, R.K. (2004) E16: The 3x + 1 Problem. Unsolved Problems in Number Theory. 3rd Edition, Springer-Verlag.
[4]
Lagarias, J.C. (2021) The 3x + 1 Problem: An Overview. https://arxiv.org/abs/2111.02635
[5]
Tao, T. (2019) Almost All Orbits of the Collatz Map Attain Almost Bounded Values. https://arxiv.org/abs/1909.03562
[6]
Carbó-Dorca, R. (2023) Collatz Conjecture Redefinition on Prime Numbers. Journal of Applied Mathematics and Physics, 11, 147-157. https://doi.org/10.4236/jamp.2023.111011
[7]
Furuta, M. (2022) Proof of Collatz Conjecture Using Division Sequence. Advances in Pure Mathematics, 12, 96-108. https://doi.org/10.4236/apm.2022.122009
[8]
Zhang, J. and Zhang, X. (2023) Collatz Iterative Trajectories of All Odd Numbers Attain Bounded Values. Journal of Applied Mathematics and Physics, 11, 3030-3041. https://doi.org/10.4236/jamp.2023.1110200
[9]
Remer, M. (2023) A Comparative Analysis of the New-3(-n)-1 Remer Conjecture and a Proof of the 3n+1 Collatz Conjecture. Journal of Applied Mathematics and Physics, 11, 2216-2220. https://doi.org/10.4236/jamp.2023.118143
[10]
Kay, D.C. (2021) Collatz Sequences and Characteristic Zero-One Strings: Progress on the 3x+1 Problem. American Journal of Computational Mathematics, 11, 226-239. https://doi.org/10.4236/ajcm.2021.113015
[11]
Kay, D.C. (2023) A Solution to the 3x + 1 Problem. American Journal of Computational Mathematics, 13, 371-377. https://doi.org/10.4236/ajcm.2023.132020
[12]
Li, K. (2022) On the Change Rule of 3x + 1 Problem. Journal of Applied Mathematics and Physics, 10, 850-864. https://doi.org/10.4236/jamp.2022.103058
[13]
Bermúdez Gómez, S. (2023) The 3x + 1 Conjecture, a Direct Path. American Journal of Computational Mathematics, 13, 350-355. https://doi.org/10.4236/ajcm.2023.132018
[14]
Hu, Z.J. (2021) The Analysis of Convergence for the 3X + 1 Problem and Crandall Conjecture for the aX + 1 Problem. Advances in Pure Mathematics, 11, 400-407.
[15]
Wang, M., Yang, Y., He, Z. and Wang, M. (2022) The Proof of the 3X + 1 Conjecture. AdvancesinPureMathematics, 12, 10-28. https://doi.org/10.4236/apm.2022.121002
[16]
Soleymanpour, R.H. (2021) A Proof of Collatz Conjecture Based on a New Tree Topology. https://arxiv.org/abs/2104.12135
