Collatz conjecture answer
Yesterday I watched this video https://youtu.be/QgzBDZwanWA about 3n + 1, I'm not a mathematician, but in my opinion, I understand how this can be proved.
for this we will have to invent a new world in mathematics.
the first thing we have to come up with is the concept of related numbers. let's say 1 this is the first generation 1 * 2 the second generation 2 * 2 the third generation
1, 2, 4, 8, 16, 32, 64................ let's call this group the first group in honor of the ancestor.
according to our logic, this whole sequence consists of relatives of even numbers, which have a common ancestor 1 of an odd number.
let's take another other tree of relatives
of natural numbers, the next tree of relatives is 3
3, 6, 12, 24, 48, 96, 192............ third group
the next tree of relatives is 5
5, 10, 20, 40, 80, 160, 320............ fifth group
Do related groups overlap with other groups?
the answer is never, the group goes to infinity and always produces unique numbers because they have different ancestors.
in reverse order, if we divide the numbers of one group by two, it always ends with an odd number, with the exception of one odd number, which is a common ancestor
64 / 2 = 32 , 32 / 2 =16 , 16 / 2 = 8 , 8 / 2 =4 , 4 / 2=2 , 2 / 2= 1
192 / 2=96 , 96 / 2=48, 48 / 2=24, 24 / 2=12, 12 / 2=6, 6 / 2= 3
320 / 2=160, 160 / 2=80, 80 / 2=40, 40 / 2=20, 20 / 2=10, 10 / 2= 5
Grandfather and grandson formula
| Results for Group 1 | Results for Group 2 | Results for Group 3 | ||||
| N*3+1 | N*3+3 | N*3+5 | ||||
| A | B | C | ||||
| 1 | 99 | 99 | 99 | 99 | 99 | 99 |
| 2 | 298 | =A1*3+1 | 300 | =B1*3+3 | 302 | =C1*3+5 |
| 3 | 149 | =IF(ISEVEN(A2),A2/2,A2*3+1) | 150 | =IF(ISEVEN(B2),B2/2,B2*3+3) | 151 | =IF(ISEVEN(C2),C2/2,C2*5+5) |
| 4 | 448 | =IF(ISEVEN(A3),A3/2,A3*3+1) | 75 | =IF(ISEVEN(B3),B3/2,B3*3+3) | 760 | =IF(ISEVEN(C3),C3/2,C3*5+5) |
| 5 | 224 | =IF(ISEVEN(A4),A4/2,A4*3+1) | 228 | =IF(ISEVEN(B4),B4/2,B4*3+3) | 380 | =IF(ISEVEN(C4),C4/2,C4*5+5) |
| 6 | 112 | =IF(ISEVEN(A5),A5/2,A5*3+1) | 114 | =IF(ISEVEN(B5),B5/2,B5*3+3) | 190 | =IF(ISEVEN(C5),C5/2,C5*5+5) |
| 7 | 56 | =IF(ISEVEN(A6),A6/2,A6*3+1) | 57 | =IF(ISEVEN(B6),B6/2,B6*3+3) | 95 | =IF(ISEVEN(C6),C6/2,C6*5+5) |
| 8 | 28 | =IF(ISEVEN(A7),A7/2,A7*3+1) | 174 | =IF(ISEVEN(B7),B7/2,B7*3+3) | 480 | =IF(ISEVEN(C7),C7/2,C7*5+5) |
| 9 | 14 | =IF(ISEVEN(A8),A8/2,A8*3+1) | 87 | =IF(ISEVEN(B8),B8/2,B8*3+3) | 240 | =IF(ISEVEN(C8),C8/2,C8*5+5) |
| 10 | 7 | =IF(ISEVEN(A9),A9/2,A9*3+1) | 264 | =IF(ISEVEN(B9),B9/2,B9*3+3) | 120 | =IF(ISEVEN(C9),C9/2,C9*5+5) |
| 11 | 22 | =IF(ISEVEN(A10),A10/2,A10*3+1) | 132 | =IF(ISEVEN(B10),B10/2,B10*3+3) | 60 | =IF(ISEVEN(C10),C10/2,C10*5+5) |
| 12 | 11 | =IF(ISEVEN(A11),A11/2,A11*3+1) | 66 | =IF(ISEVEN(B11),B11/2,B11*3+3) | 30 | =IF(ISEVEN(C11),C11/2,C11*5+5) |
| 13 | 34 | =IF(ISEVEN(A12),A12/2,A12*3+1) | 33 | =IF(ISEVEN(B12),B12/2,B12*3+3) | 15 | =IF(ISEVEN(C12),C12/2,C12*5+5) |
| 14 | 17 | =IF(ISEVEN(A13),A13/2,A13*3+1) | 102 | =IF(ISEVEN(B13),B13/2,B13*3+3) | 80 | =IF(ISEVEN(C13),C13/2,C13*5+5) |
| 15 | 52 | =IF(ISEVEN(A14),A14/2,A14*3+1) | 51 | =IF(ISEVEN(B14),B14/2,B14*3+3) | 40 | =IF(ISEVEN(C14),C14/2,C14*5+5) |
| 16 | 26 | =IF(ISEVEN(A15),A15/2,A15*3+1) | 156 | =IF(ISEVEN(B15),B15/2,B15*3+3) | 20 | =IF(ISEVEN(C15),C15/2,C15*5+5) |
| 17 | 13 | =IF(ISEVEN(A16),A16/2,A16*3+1) | 78 | =IF(ISEVEN(B16),B16/2,B16*3+3) | 10 | =IF(ISEVEN(C16),C16/2,C16*5+5) |
| 18 | 40 | =IF(ISEVEN(A17),A17/2,A17*3+1) | 39 | =IF(ISEVEN(B17),B17/2,B17*3+3) | 5 | =IF(ISEVEN(C17),C17/2,C17*5+5) |
| 19 | 20 | =IF(ISEVEN(A18),A18/2,A18*3+1) | 120 | =IF(ISEVEN(B18),B18/2,B18*3+3) | ||
| 20 | 10 | =IF(ISEVEN(A19),A19/2,A19*3+1) | 60 | =IF(ISEVEN(B19),B19/2,B19*3+3) | ||
| 21 | 5 | =IF(ISEVEN(A20),A20/2,A20*3+1) | 30 | =IF(ISEVEN(B20),B20/2,B20*3+3) | ||
| 22 | 16 | =IF(ISEVEN(A21),A21/2,A21*3+1) | 15 | =IF(ISEVEN(B21),B21/2,B21*3+3) | ||
| 23 | 8 | =IF(ISEVEN(A22),A22/2,A22*3+1) | 48 | =IF(ISEVEN(B22),B22/2,B22*3+3) | ||
| 24 | 4 | =IF(ISEVEN(A23),A23/2,A23*3+1) | 24 | =IF(ISEVEN(B23),B23/2,B23*3+3) | ||
| 25 | 2 | =IF(ISEVEN(A24),A24/2,A24*3+1) | 12 | =IF(ISEVEN(B24),B24/2,B24*3+3) | ||
| 26 | 1 | =IF(ISEVEN(A25),A25/2,A25*3+1) | 6 | =IF(ISEVEN(B25),B25/2,B25*3+3) | ||
| 27 | 3 | =IF(ISEVEN(B26),B26/2,B26*3+3) | ||||