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.
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)|