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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/4013334#.X1DhOcgzbIU).

- Eduard Dyachenko (talk) 08:59, 27 March 2020 (UTC) E.Dyachenko (dyachenko.eduard@gmail.com)[reply]