Lothar Otto Collatz conjectured that

for any integer n,

- when n is even, then divide by 2 (
**n ⇒ n/2**) - when n is odd, then multiply by 3 and add 1 (
**n ⇒ 3n+1**)

n always converges to 1.

all n ≤ 2*10^{15} was checked
(For current situation, please check here).

Sometimes n becomes very large during the iteration of computation.

Let max(n) be the maximun value of n during its iterations. Then it seems to be,

**
max(n) ≤ n ^{a}, a=2+ε (conjecture)
**

If this conjecture is right, then any n has a finite upper bound for its iteration,

so we can prove that any n does not divergent to infinity.

(But there still remains a possibility of a loop.)

In order to understand the difficulty of this problem,

please try to challenge following questions.

- Q1 : Does 27 converge to 1 ?
- Q2 : How about 447 ?
- Q3 : Can you construct the number which converges after n times iterations ?

Chapter 6 Additive Palindromicness of Natural Numbers |
"Mathematician's Secret Room" | Chapter 8 Continued Fraction and Pell's Equation |
---|---|---|

Chapter 6 (Japanese) | index (Japanese) | Chapter 8 (Japanese) |

E-mail : kc2h-msm@asahi-net.or.jpHisanori Mishima