File talk:Collatz5.svg
This image really needs some better labeling. As it is, it's pretty unclear what exactly this graph is meant to represent. It doesn't help that the Collatz article has no caption for this and no description of it in text.TV4Fun (talk) 17:27, 7 March 2010 (UTC)
About the combination of: harmonic intervals built on the 'tetrada', the problem 3x+1 and the fundamental constant 4:3
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 ([email protected])
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 ([email protected])