Chapter 7 : Collatz's Conjecture (E16 The 3x+1 problem)

(First : June 04, 2001; update : April 12, 2008) [Japanese]


Lothar Otto Collatz conjectured that

for any integer n,

n always converges to 1.

all n ≤ 2*1015 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) ≤ na,   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.


  1. Definition and program
  2. Plan for search
  3. Omittable n for search
  4. Relation between n and max(n)

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 :
Hisanori Mishima