The 3x +1 (or Collatz) sequences and their properties ... SUMMARY: The so-called 3x +1 problem is to prove that all 3x +1 sequences eventually converge. ...

... a new approach to the 3x + 1 Problem , an approach based on two remarkably simple structures that underlie the 3x + 1 function. ...

Despite this doleful pronouncement, study of the 3x +1 problem has not been without reward. ... A Continuous Extension of the 3x +1 Problem to the Real Line , Dynamics of ...

This is equivalent to the (3n1 ) problem on positive integers. ... The 3x + 1 problem and its generalizations, American Mathematical Monthly Volume ...

Mathematicians who refer to the problem as the 3x +1 problem were never brainwashed by FORTRAN (as I was) into the belief that n, not x, stands for an integer. ...

1 Problem , the Syracuse Problem , the Collatz Problem , Ulam's Problem , ... Other equivalent formulations of the 3x + 1 problem are given in the literature; we base our ...

Density Bounds for the 3x +1 Problem I. Tree-Search Method , ... Density Bounds for the 3x +1 Problem II. Krasikov Inequalities . David Applegate and ...

Contents Next: Introduction. The 3x + 1 Problem and its Generalizations ... Generalizations of problem . Algorithmic decidability questions. Existence of stopping ...

The 3x +1 Problem is perhaps today's most enigmatic unsolved mathematical ... The problem is also referred to as the 3n + 1 problem and is associated with ...

The 3x +1 problem is one I find interesting as it is based on some ... The 3x +1 equation appears to connect all the positive integers into a tree rooted on the ...