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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 on the integers, 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 $m\geq1$ has some iterate $T^{(k)}(m)=1$." Section 1 restates it on p. 1 in the same terms for every $m\ge1$. The abstract records that the conjecture remains unsolved, and §1 repeats this on p. 2, adding that the proofs claimed by Yamada (1981), Cadogan (2006) and Bruckman (2008) are incomplete.

Reported verification (p. 1). Section 1 reports that the conjecture has been verified up to $17\times2^{58}>4.899\times10^{18}$, as of Feb. 21, 2008, by the ongoing computation of T. Oliveira e Silva (2004+), and that an independent computation of Roosendaal (2004+) verifies it up to $612\times2^{50}>6.89\times10^{17}$. The bibliography reports these bounds and does not check them.

Source. Jeffrey C. Lagarias, The 3x+1 Problem: An Annotated Bibliography, II (2000-2009), arXiv:math/0608208v6, p. 1 (the abstract and the opening of §1, which runs to p. 2) and p. 2, read on the page images. The edition read is identified on the source card.

Read depth. Claims checked: the two definitions, the conjecture and the reported bounds were read clause by clause on the page images of pp. 1--2. 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, reports computations that check it on an initial range, records it as unsolved, and proves nothing about it.