3x+1 Minus the +

The Structure of the 3x + 1 Function: An Introduction

THE CONGRUENCE STRUCTURE OF THE 3x + 1 MAP

Let T : Z→ Z be defined by T (x) = 12x if x is even, and T (x) = 1 2 (3x + 1) if x is odd. The 3x + 1 Conjecture asserts that every positive x has an iterate T(x) = 1. It is known that T maps congruence classes with modulus 2 to those with modulus equal to a power of 3. We describe properties of the image class residues and use those properties to show that, under iteration by T , any congruenc...

Lower bounds for the total stopping time of 3x + 1 iterates

The total stopping time σ∞(n) of a positive integer n is the minimal number of iterates of the 3x + 1 function needed to reach the value 1, and is +∞ if no iterate of n reaches 1. It is shown that there are infinitely many positive integers n having a finite total stopping time σ∞(n) such that σ∞(n) > 6.14316 log n. The proof involves a search of 3x+1 trees to depth 60, A heuristic argument sug...

The Ergodicity of the Collatz Process in Positive Integer Field

Since 1930s, researchers have deeply investigated the 3x+1 problem. Until now, the 3x+1 problem has obtained many names, such as the Kakutani’s problem, the Syracuse problem and Ulam’s problem and so on[11, 9]. The 3x+ 1 problem can be stated from the viewpoint of computer algorithm as follows: For any given integer x, if x is odd, then let x := 3x + 1; if x is even, then let x := x/2; if we re...

ar X iv : m at h / 04 11 14 0 v 2 [ m at h . N T ] 4 M ay 2 00 5 THE 3 x + 1 SEMIGROUP

The 3x + 1 semigroup is the multiplicative semigroup S of positive rational numbers generated by { 2k+1 3k+2 : k ≥ 0} together with {2}. This semigroup encodes backwards iteration under the 3x + 1 map, and the 3x + 1 conjecture implies that it contains every positive integer. This semigroup is proved to be the set of positive rationals a b in lowest terms with b 6≡ 0( mod 3), and so contains al...