File talk:Collatz5.svg
About the combination of: harmonic intervals built on the 'tetrada', the problem 3x+1 and the fundamental constant 4:3
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A. The following work DOI 10.5281/zenodo.3630682 proves that the Collatz transformation leads the “length of a number” indicated in the system q = 4∩3 to a unit length.
- The main idea is that reduction of the "length of the number" occurs when converting oddness of the form 4k+1 and preservation of the "length of the number" when converting oddness of the form 4k+3.
- Since the transformation 4k+3 cannot be stored indefinitely, periodically the "length of the number" decreases.
- As a result of consequent iterations the number transforms into the form 2^p/3^q .
B. Simultaneously, during the Collatz transformation of the number (position A), it appears in the system qi = 2∩3 .
C. The record of the number in system qi = 2∩3 (B) consists of the sum of elementary numbers in form {(2^a(ⅈ) -2^b(ⅈ) )/3^ }
D. Example: number 27 = 3/3⋅q^9+2/3⋅q^8+1/3 q^7+0⋅q^6+2/3 q^5+0⋅q^4+1/3 q^3+1/3 q^2+0⋅q^1+1/3 q^0 and its transformation (see table)
| i | n(i) | r(i) | -r(i)>1 | 4k(i)+1 | 4k(i)+3 | k(i) | P(i) | |
|---|---|---|---|---|---|---|---|---|
| 0 | 27 | -1 | 4*6+3 | 14+13 | even | 1 | ||
| 1 | 41 | -2 | 2 | 4*10 + 1 | ||||
| 2 | 31 | -1 | 4*7+3 | 16+15 | odd | 4 | ||
| 3 | 47 | -1 | 4*11+3 | odd | ||||
| 4 | 71 | -1 | 4*17+3 | odd | ||||
| 5 | 107 | -1 | 4*26+3 | even | ||||
| 6 | 161 | -2 | 2 | 4*40+1 | ||||
| 7 | 121 | -2 | 2 | 4*30+1 | ||||
| 8 | 91 | -1 | 4*22+3 | 46+45 | even | 1 | ||
| 9 | 137 | -2 | 2 | 4*34+1 | ||||
| 10 | 103 | -1 | 4*25+3 | 52+51 | odd | 2 | ||
| 11 | 155 | -1 | 4*38+3 | even | ||||
| 12 | 233 | -2 | 2 | 4*58+1 | ||||
| 13 | 175 | -1 | 4*43+3 | 88+87 | odd | 3 | ||
| 14 | 263 | -1 | 4*65+3 | odd | ||||
| 15 | 395 | -1 | 4*98+3 | even | ||||
| 16 | 593 | -2 | 2 | 4*148+1 | ||||
| 17 | 445 | -3 | 3 | 4*111+1 | ||||
| 18 | 167 | -1 | 4*41+3 | 84+83 | odd | 2 | ||
| 19 | 251 | -1 | 4*62+3 | even | ||||
| 20 | 377 | -2 | 2 | 4*94+1 | ||||
| 21 | 283 | -1 | 4*70+3 | 213+212 | even | 1 | ||
| 22 | 425 | -2 | 2 | 4*106+1 | ||||
| 23 | 319 | -1 | 4*79+3 | 160+159 | odd | 5 | ||
| 24 | 479 | -1 | 4*119+3 | odd | ||||
| 25 | 719 | -1 | 4*179+3 | odd | ||||
| 26 | 1079 | -1 | 4*269+3 | odd | ||||
| 27 | 1619 | -1 | 4*404+3 | even | ||||
| 28 | 2429 | -3 | 3 | 4*607+1 | ||||
| 29 | 911 | -1 | 4*227+3 | 456+455 | odd | 3 | ||
| 30 | 1367 | -1 | 4*341+3 | odd | ||||
| 31 | 2051 | -1 | 4*512+3 | even | ||||
| 32 | 3077 | -4 | 4 | 4*769+1 | ||||
| 33 | 577 | -2 | 2 | 4*144+1 | ||||
| 34 | 433 | -2 | 2 | 4*108+1 | ||||
| 35 | 325 | -4 | 4 | 4*81+1 | ||||
| 36 | 61 | -3 | 3 | 4*15+1 | ||||
| 37 | 23 | -1 | 4*5+3 | 12+11 | odd | 2 | ||
| 38 | 35 | -1 | 4*8+3 | even | ||||
| 39 | 53 | -5 | 5 | 4*13+1 | ||||
| 40 | 5 | -4 | 4 | 4*1+1 | ||||
| 41 | 1 | 0 | ||||||
| balance | -70 | 46 | 24 |
E. Find complete prove here (https://zenodo.org/record/3630682#.XjagkGhKjIU).
- Eduard Dyachenko (talk) 08:59, 27 March 2020 (UTC) E.Dyachenko (dyachenko.eduard@gmail.com)
I believe the table, posted along with the graph, is a worthy visualization how graph behaves.
This is a supporting element providing explanation to the picks of the graph.
E.Dyachenko (dyachenko.eduard@gmail.com)
— Preceding unsigned comment was added by 93.141.73.117 (talk) 14:10, 1 April 2020 (UTC)
- Not sure if this analysis is related particularly to the image, probably it should be presented elsewhere. —Mykhal (留言) 18:42, 26 August 2022 (UTC)
Graph type
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@Pokipsy76: Is there a reason for this graph to be a continous line? Probably it should be set of discrete points. Regards, —Mykhal (留言) 18:44, 26 August 2022 (UTC)
