It has been conjectured that all integers eventually yield a 1 . The "Collatz problem ", also known as the "3x +1 ?? problem , is to determine whether the conjecture is true. ...

Collatz (3n+1 , 3x +1 )-problem . Besides some statistical extrapolations and some sportive ... But they gave new insight in the problem in a way, that I didn't see in the internet ...

1 ,1 . The 3x +1 or Collatz problem is as follows: start with any number n. If n is even, divide it by 2, otherwise multiply it by 3 and add 1 . Do we always reach 1 ? ...

In the `3x +1 ' problem , these values for the starting value set new records for number of ... Both the 3x +1 steps and the halving steps are counted. B. Hayes, ...

????? ?? ?????? 3x +1 problem . ???????? ???? ???? ???????? ????? ?? ... 1 , 1 . The length of an odd block is the number of odd numbers found within it > ...

Recently, he has announced a solution to the famous 3x +1 problem . ... Cadogan, Charles, Trajectories in the 3x +1 Problem . Journal of Combinatorial Mathematics and Combinatorial ...

Just like the 3x +1 function, it appears that, starting from any ... [1 ] J. C. Lagarias and A. Weiss, The 3x +1 problem : two stochastic models, The Annals of Applied Probability, ...

We can represent the "3x +1 problem " as a tree, in which the leaves ... Figure 1 : "3x +1 Tree" In that tree the leaves are connected downstream by one of the ...

Every problem is a possible seed for new problems , which come from changing, adding to, ... example, the famous 3x + 1 conjecture explores the fate of sequences ...

The 3x +1 Problem : An Annotated Bibliography (1963-1999) , Jeffrey C. Lagarias, Unpublished . ... Lower bounds for the total stopping time of 3x +1 iterates , ...