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Statement

The two maps (p. 1). The $3x+1$ function is the map $T:\mathbb Z\to\mathbb Z$ with $T(x)=(3x+1)/2$ for $x\equiv1\pmod2$ and $T(x)=x/2$ for $x\equiv0\pmod2$ (abstract, and again in §1). The Collatz function, defined in §1, is $C(x)=3x+1$ for $x\equiv1\pmod2$ and $C(x)=x/2$ for $x\equiv0\pmod2$; the bibliography states the $3x+1$ problem, or Collatz problem, as the task of proving that from any positive integer some iterate of $C$ takes the value $1$.

$3x+1$ Conjecture (p. 1, unnumbered), quoted from the abstract: "The $3x+1$ Conjecture asserts that each $m\ge1$ has some iterate $T^{(k)}(m)=1$." Section 1 restates it on p. 1 in the same terms for every $m\ge1$. The abstract adds that the conjecture remains unsolved, as of the version read (dated January 1, 2011).

Source. Jeffrey C. Lagarias, The 3x+1 Problem: An Annotated Bibliography (1963-1999) (Sorted by Author), arXiv:math/0309224v13, p. 1 (the abstract and the opening of §1, which runs to p. 2), read on the page image. The edition read is identified on the source card.

Read depth. Claims checked: the two definitions and the conjecture were read clause by clause on the page image of p. 1. The conjecture is open; the bibliography proves nothing about it, and nothing here is independently reviewed.

Proof pointer

None: the statement is a conjecture, and the bibliography is a list of annotated works, not a research paper.

Dependencies

None.

Bears on

• Problem 1135: the problem's map $f$ is the bibliography's $T$ restricted to the positive integers, and its question, whether every $m\ge1$ has some $k\ge1$ with $f^{(k)}(m)=1$, is the bibliography's $3x+1$ Conjecture. The print does not restrict $k$; the two readings agree, since for $m\ge2$ the iterate $T^{(0)}(m)=m$ is not $1$, and for $m=1$ one has $T(1)=2$ and $T^{(2)}(1)=1$ (a remark of this page). The bibliography states the conjecture, records it as unsolved, and proves nothing about it.