# Supplement to the Collatz Conjecture

## DOI:

https://doi.org/10.24297/jam.v15i0.7876## Keywords:

sequences## Abstract

For any natural number was created the supplement sequence, that is convergent together with the original sequence. The parameter - index was defined, that is the same tor both sequences. This new method provides the following results:

- All natural numbers were distributed into different classes according to the corresponding indexes;
- The analytic formulas ( not by computer performed routine calculations) were produced, the formulas for groups of consecutive natural numbers of different lengths, having the same index;
- The new algorithm to find index for any natural number was constructed and proved.

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## References

1. Maddux, Cleborne D.; Johnson, D. Lamont (1997). Logo: A Retrospective. New York: Haworth Press. p. 160. ISBN 0-7890-0374-0. The problem is also known by several other names, including: Ulam's conjecture, the Hailstone problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and the Collatz problem.

2. Lagarias, Jeffrey C. (1985). "The 3x + 1 problem and its generalizations". The American Mathematical Monthly. 92 (1): 3–23. JSTOR 2322189.

3. Pickover, Clifford A. (2001). Wonders of Numbers. Oxford: Oxford University Press. pp. 116–118. ISBN 0-19-513342-0.

4. Guy, Richard K. (2004). ""E17: Permutation Sequences"". Unsolved problems in number theory (3rd ed.). Springer-Verlag. pp. 336–7. ISBN Zbl 1058.11001.

5. Guy, R. K. (1983). "Don't try to solve these problems". Amer. Math. Monthly. 90: 35–41. doi:10.2307/2975688. JSTOR 2975688. By this Erdos means that there aren't powerful tools for manipulating such objects.

6. Lagarias, Jeffrey C., ed. (2010). The ultimate challenge: the 3x+1 problem. Providence, R.I.: American Mathematical Society. p. 4. ISBN 0821849409.

7. Leavens, Gary T.; Vermeulen, Mike (December 1992). "3x+1 Search Programs". Computers & Mathematics with Applications. 24 (11): 79–99. doi:10.1016/0898-1221(92)90034-F.

8. Roosendaal, Eric. "3x+1 Delay Records". Retrieved 30 June 2017. (Note: "Delay records" are total stopping time records.)

9. Kurtz, Stuart A.; Simon, Janos (2007). "The Undecidability of the Generalized Collatz Problem". In Cai, J.-Y.; Cooper, S. B.; Zhu, H. Proceedings of the 4th International Conference on Theory and Applications of Models of Computation, TAMC 2007, held in Shanghai, China in May 2007. pp. 542–553. doi:10.1007/978-3-540-72504-6_49. ISBN 3-540-72503-2. As PDF.

2. Lagarias, Jeffrey C. (1985). "The 3x + 1 problem and its generalizations". The American Mathematical Monthly. 92 (1): 3–23. JSTOR 2322189.

3. Pickover, Clifford A. (2001). Wonders of Numbers. Oxford: Oxford University Press. pp. 116–118. ISBN 0-19-513342-0.

4. Guy, Richard K. (2004). ""E17: Permutation Sequences"". Unsolved problems in number theory (3rd ed.). Springer-Verlag. pp. 336–7. ISBN Zbl 1058.11001.

5. Guy, R. K. (1983). "Don't try to solve these problems". Amer. Math. Monthly. 90: 35–41. doi:10.2307/2975688. JSTOR 2975688. By this Erdos means that there aren't powerful tools for manipulating such objects.

6. Lagarias, Jeffrey C., ed. (2010). The ultimate challenge: the 3x+1 problem. Providence, R.I.: American Mathematical Society. p. 4. ISBN 0821849409.

7. Leavens, Gary T.; Vermeulen, Mike (December 1992). "3x+1 Search Programs". Computers & Mathematics with Applications. 24 (11): 79–99. doi:10.1016/0898-1221(92)90034-F.

8. Roosendaal, Eric. "3x+1 Delay Records". Retrieved 30 June 2017. (Note: "Delay records" are total stopping time records.)

9. Kurtz, Stuart A.; Simon, Janos (2007). "The Undecidability of the Generalized Collatz Problem". In Cai, J.-Y.; Cooper, S. B.; Zhu, H. Proceedings of the 4th International Conference on Theory and Applications of Models of Computation, TAMC 2007, held in Shanghai, China in May 2007. pp. 542–553. doi:10.1007/978-3-540-72504-6_49. ISBN 3-540-72503-2. As PDF.

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## Published

2018-12-01

## How to Cite

*JOURNAL OF ADVANCES IN MATHEMATICS*,

*15*, 8120–8132. https://doi.org/10.24297/jam.v15i0.7876

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