Sergey Zhestkov, a lecturer at MIPT and concurrently an OTUS expert, invites everyone to a free demo lesson of the advanced course "Mathematics for Data Science" , on the topic: "Maps, their matrix and diagonalization . "
And we traditionally share with you the translation of interesting material.
Despite recent sympathies with the infamous Collatz hypothesis, we still can't figure out if a number can get out of an infinite loop.
This article comes with a warning: don't try to solve this math problem.
. , . , : , ; , 3 1. . , . , .
, , 10: 10 - , 5. 5 - , 3 1. 16, , 2 8, 8 4, 2, , 1. 1 , 1. 4, , : 4 2, 1, 4, . .
11: , 1. 34, , 17, 1, 52, , 26, , 13, 1, 40, , 20, 10, 5, 1, 16, , 8, 4, 2 1. .
, , . , , : , . , , .
, . , . : , - , , , . , .
, :
(even - , odd - )
«» : n , , n . f , : , f (10) = 10/2 = 5, 10 , f (5) = 3 × 5 + 1 = 16, 5 . 3n + 1.
«» f. - , , - , . «» . 10 f, :
f (10) = 10/2 = 5
f (5) = 3 × 5 + 1 = 16
f (16) = 16/2 = 8
f (8) = 8/2 = 4
. 10 f:
10 → 5 → 16 → 8 → 4 → 2 → 1 → 4 → 2 → 1 → …
, 1 → 4 → 2 → 1 →….
, 11 f
11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1 → 4 → ….
. , , 4 → 2 → 1 →…. 9 19 , , 27. , 111 .
, f 1. , 26⁸. , , 300 . ( !)
, : , 1. , , ℊ.
ℊ f, 1 , . ℊ f , . , 10 11 ℊ:
10 → 5 → 6 → 3 → 4 → 2 → 1 → 2 → 1 → 2 → …
11 → 12 → 6 → 3 → 4 → 2 → 1 → 2 → 1→ 2 → …
, 11 1 ℊ, f. 27 1 ℊ.
27 → 28 → 14 → 7 → 8 → 4 → 2 → 1 → 2 → …
ℊ , f, :
→ 2 → 1 → 2 → 1 → ….
, ℊ 1. «», n + 1. , , , - - 26⁸ - , . , . .
-, , . , n , ℊ(n) = n/ 2 < n. , , .
, n , ℊ(n) = n + 1, n. , + 1 , : ℊ + 1 . n :
,
.
- ,
, n. , > 1,
, ℊ 1, . : , , . - 1 . 1, , .
? .
ℊ, f . ℊ, f , , , : f . f, n :
. , n:
, n. , , . .
- , , , , : , , 1? , , , , : ,
.
, 3n + 1 . 3n + 1 4,
, . 3n + 1 4,
, . , , , .
. , 50%
,
. n > 1 , n, . 50%- ,
, , 25%- , . . , , . , , . , .
, « » . , , , , , 1, , , . 1976 - , , . , , , - , 1.
,
,
( n), f(n), f(x) - , , , . , , . , , « .»
. , . , : .
1. , , 1.
2. « » n - , , n 1. , 10 6, 11 14. 5.
3. :
, 1 → 2 → 1 → 2 → 1… . ?
, 1:
, 1. ,
, , 1.
, 2:
2^5 5,
…. 2^4 4, , 2^4, 5. , 5 → 16 → 8 → 4 → 2 → 1. ?
, 3:
:
5 → 14 → 7 → 20 → 10 → 5 → …
17 → 50 → 25 → 74 → 37 → 110 → 55 → 164 → 82 → 41 → 122 → 61 → 182 → 91 → 272 → 136 → 68 → 34 → 17 → …