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Statement

Section 5, "Cycles", pp. 15--17 of the English version. Throughout, $T$ is the compressed map of p. 2 and a cycle is a periodic orbit of $T$.

The author's observations (pp. 15--16; credited to the survey's [21], an announcement by the author at the 1999 Eichstätt conference, and for the identity also to Monks [57], 2002). For a cycle $\Omega$ of $T$ with odd terms $\Omega_{odd}$ and even terms $\Omega_{even}$, summing $\sum_{x\in\Omega}x=\sum_{x\in\Omega}T(x)$ gives

$$\sum_{x\in\Omega_{even}}x=\sum_{x\in\Omega_{odd}}x+|\Omega_{odd}|.$$

From the action of $T$ on residues mod $3$ and mod $4$ (the survey's Figure 3), no integer cycle other than $\{0\}$ has an element divisible by $3$, and in any cycle the number of terms congruent to $1$ mod $4$ equals the number congruent to $2$ mod $4$.

Circuits (p. 16). A circuit is a cycle consisting of $k$ odd elements followed by $l$ even ones. Davison ([27], 1976) put circuits in one-to-one correspondence with the positive integer solutions $(k,l,h)$ of $(2^{k+l}-3^k)h=2^l-1$, the survey's equation (1); by continued fractions and transcendence theory (Steiner [71], 1977; Rozier [66], 1990) its only solution is $(1,1,1)$, so $\{1,2\}$ is the only circuit.

Cycle lengths (pp. 16--17). For a nontrivial cycle $\Omega$ of $T$ with smallest term $m$, largest term $M$ and $|\Omega_o|$ odd terms, Eliahou ([30], 1993) proved $\log_2(3+1/M)\le|\Omega|/|\Omega_o|\le\log_2(3+1/m)$, the survey's (2), and with the bound $m>2^{40}$ and the Diophantine approximation of $\log_23$ showed $|\Omega|=301994a+17087915b+85137581c$ with $a,b,c$ nonnegative integers, $b\ge1$ and $ac=0$. Tempkin and Arteaga ([75], 1997, a draft) tightened (2) and used a better lower bound on $m$ to obtain

$$|\Omega|=187363077a+272500658b+357638239c,$$

with $a,b,c$ nonnegative integers, $b\ge1$ and $ac=0$; since $b\ge1$, a nontrivial cycle has at least $272{,}500{,}658$ terms, the record of Section 2 (p. 3).

Brox (p. 17). With $\sigma_i$ the number of terms of a cycle congruent to $i$ mod $4$, Brox ([17], 2000) proved that only finitely many cycles satisfy $\sigma_1<2\log(\sigma_1+\sigma_3)$.

Source. M. Chamberland, An Update on the $3x+1$ Problem, author's English version of the survey in Butll. Soc. Catalana Mat. 18 (2003), 19--45; pp. 15--17 of the English version, read on the page images. The survey names the odd terms $\Omega_0$ in the sentence before (2) and writes $\Omega_o$ in (2) itself; this page uses $\Omega_o$. The edition read is identified on the source card.

Read depth. Claims checked: the passage was read clause by clause on the page images. Apart from the author's own observations, these are a survey's reports of other authors' results; the cited sources were not read here.

Proof pointer

The cycle identity: sum $T(x)=x/2$ over the even terms and $T(x)=(3x+1)/2$ over the odd terms, set the total equal to $\sum_{x\in\Omega}x$ and multiply by $2$ (a remark of this page). The other results: Davison, Proc. Sixth Manitoba Conf. Numer. Math. (1976), 155--159; Eliahou, Discrete Math. 118 (1993), 45--56; Brox, Acta Arith. 92 (2000), 181--188; Tempkin and Arteaga's 1997 draft; none of them held.

Dependencies

The cited papers, as reported.

Bears on

• Problem 1135: a negative answer to the problem's question would need an orbit of $f$ (the survey's $T$) that is divergent or ends in a nontrivial cycle; these results restrict the second alternative (no nontrivial circuit, no nontrivial cycle of fewer than $272{,}500{,}658$ terms) without excluding it. Partial results only; the current cycle-exclusion frontier is Hercher's.