The 3x+1 conjecture

Let $\Pi$ be the set of positive integers relatively prime to $2$ and $3$ . For $x\in\Pi$ , define the $(3x+1)$ -Map $M$ by choosing $k\geq 1$ such that $2^k\parallel(3x+1)$ and setting

y=\frac{3x+1}{2^k}.

$(3x+1)$ -Conjecture. For every $x\in\Pi$ , there is an integer $n$ such that

M^n(x)=1.

This is the classical Collatz-type assertion that every positive integer in the natural domain of the odd-step map eventually reaches the fixed point $1$ . The source reports computational verification for all $x\in\Pi$ with $0<x<2^{60}$ , but the universal statement remains open.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The $(3x+1)$ -conjecture
Let $\Pi=6\mathbb{Z}^{+}+\{1,5\}$ be the set of positive integers not divisible by $2$ or $3$ . For $x\in\Pi$ , define $T(x)$ by choosing the unique integer $k>0$ such that
$T(x)=\frac{3x+1}{2^k}$
is odd.

$(3x+1)$ -conjecture. For every $x\in\Pi$ , there is an integer $n$ such that
$T^n(x)=1.$
This is the classical Collatz-type conjecture for the $(3x+1)$ -map. The source notes that the claim has been verified computationally for all integers $x$ with $0<x<2^{60}$ , while its validity for every $x\in\Pi$ remains open.
source: Alex V. Kontorovich and Yakov G. Sinai, “Structure Theorem for (d,g,h)-Maps”, arXiv:math/0601622 (2006).

Sources & referencesView supporting material

Primary source

Alex V. Kontorovich and Steven J. Miller, “Benford's Law, Values of L-functions and the 3x+1 Problem”, arXiv:math/0412003 (2005).

1 week ago

The 3x+1 conjecture

Equivalent formulations 1