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Statement

Section 8, "Future Prospects" (p. 20), lists (C1)--(C5) as samples of the "easier" problems that the analysis of the $3x+1$ problem has produced and that it says remain unsolved; the two below concern the $3x+1$ function $T$ itself ((C3) concerns the $5x+1$ function, (C4) and (C5) permutations given by generalized Collatz functions).

(C1) (Finite Cycles Conjecture), quoted: "Does the $3x+1$ function have finitely many cycles (i.e. finitely many purely periodic orbits on the integers)? This is conjectured to be the case."

(C2) (Divergent Trajectories Conjecture-1), quoted: "Does the $3x+1$ function have a divergent trajectory, i.e., an integer starting value whose iterates are unbounded? This is conjectured not to be the case."

The open problem on $\pi_1$ (p. 20, unlabeled). Let $\pi_1(x)$ be the number of integers less than $x$ that reach $1$ under the $3x+1$ iteration. The survey recalls that by Krasikov and Lagarias (its [57]) there is a positive constant $c_0$ with $\pi_1(x)>c_0x^{0.84}$, and states as open the problem of showing that for each $\epsilon>0$ there is a positive constant $c(\epsilon)$ with $\pi_1(x)>c(\epsilon)x^{1-\epsilon}$. (The same paper is cited for (W5) of Section 6 in the form "at least $X^{0.84}$, for all sufficiently large $X$".)

Source. J. C. Lagarias, The $3x+1$ problem: an overview, in The Ultimate Challenge: The $3x+1$ Problem (AMS, 2010), 3--29; the arXiv:2111.02635v1 copy, p. 20, read on the page image. The edition read is identified on the source card.

Read depth. Claims checked: the statements were read clause by clause on the page image. They are open problems; nothing here is independently reviewed.

Proof pointer

None: the statements are open. The bound $\pi_1(x)>c_0x^{0.84}$ is Krasikov and Lagarias, Acta Arith. 109 (2003), 237--258, whose source card is krasikov_lagarias_2003_bounds_difference_inequalities.

Dependencies

Krasikov and Lagarias 2003 (the survey's [57]), as reported.

Bears on

• Problem 1135: (C1) and (C2) range over all integers, the problem over the positive integers. An affirmative answer to the problem would show that on the positive integers $\{1,2\}$ is the only cycle of $T$ and no trajectory is unbounded, which is the positive-integer part of (C1) and (C2); it would not settle either as printed, and neither conjecture, nor both together, would answer the problem. An affirmative answer would make $\pi_1(x)$ count every positive integer below $x$, so the open problem on $\pi_1$ is a weak consequence of it; the survey proves none of these.