The Collatz $3x-1$ problem (the $3n+1$ problem for negative start values): exactly three linearly independent holomorphic solutions of the Berg–Opfer functional equations

For the Collatz iteration restricted to negative start values — written on positive integers as $s(n)=n/2$ when $n$ is even and $s(n)=(3n-1)/2$ when $n$ is odd — exactly three cycles are known: $S_1=\{1\}$, $S_2=\{5,7,10\}$, and $S_3=\{17,25,37,55,82,41,61,91,136,68,34\}$, and all positive integers up to $10^8$ empirically fall into one of their basins. Berg and Opfer (2013) recast the conjecture that every positive integer's trajectory enters one of these three cycles as a complex-analytic statement: the conjecture holds if and only if there are exactly three linearly independent holomorphic functions on the unit disk with Taylor coefficients in $\{0,1\}$ satisfying $h_n=h_{s(n)}$, equivalently satisfying a specific pair of functional equations; a fourth linearly independent solution would correspond to an unknown cycle or a divergent trajectory. The paper's companion conjecture (its Conjecture 2.4) — that the three basin generating functions $\eta_1,\eta_2,\eta_3$ have the unit circle as a natural boundary — was subsequently proved by Bell and Lagarias (2015) via the Pólya–Carlson theorem. The surviving open core is the three-dimensionality statement itself, equivalent to the $3x-1$ conjecture, which remains open: prove it, or refute it by exhibiting a positive integer whose trajectory enters a new cycle or diverges.

Background

The Collatz function is the map $c(n)=n/2$ if $n$ is even and $c(n)=(3n+1)/2$ if $n$ is odd, iterated from a start value $n$. For positive start values the famous Collatz conjecture asserts that every trajectory reaches $1$; it is open. For negative start values the problem is equivalent, via the change of sign $n\mapsto -n$, to iterating $s(n)=n/2$ for even $n$ and $s(n)=(3n-1)/2$ for odd $n$ on the positive integers (the $3x-1$ problem). Its behavior differs from the positive case: exactly three cycles are known, $S_1=\{1\}$, $S_2=\{5,7,10\}$, and $S_3=\{17,25,37,55,82,41,61,91,136,68,34\}$, and computations (Berg and Opfer, up to $10^8$) show every start value tested lands in one of the three basins $N_j$, with roughly equal frequencies of about one third each. The expected log-change per odd step is $\log(3/4)<0$, the same negative drift as in the $3x+1$ case, so heuristics also predict convergence to one of the three cycles; but no proof is known, and no additional cycle or divergent trajectory has been found.

Meinardus (1987) and Berg and Meinardus (1994, 1995) placed the positive Collatz problem into complex analysis; the negative-start-value problem considered here is treated analogously: one studies power series $h(z)=\sum_{n\ge 1}h_n z^n$ with coefficients in $\{0,1\}$ that are constant along trajectories, $h_n=h_{s(n)}$ for all $n\ge 1$. Such series are holomorphic (complex differentiable) on the open unit disk $D=\{z:|z|<1\}$ because their coefficients are bounded, and the coefficient condition is equivalent to a pair of functional equations for $h$, namely $h(z)+h(-z)=2h(z^2)$ and

$$h(z^3)-h(-z^3)=\frac{2z}{3}\sum_{\nu=0}^{2}\lambda^{-\nu}h(\lambda^{\nu}z^2),\qquad \lambda=e^{2\pi i/3}.$$

Solutions are built from merge classes — two integers are equivalent if their trajectories eventually coincide, and $h$ must be constant on each class. The generating function of each basin, $\eta_j(z)=\sum_{n\in N_j}z^n$ (where $N_j$ is the set of start values whose trajectory meets $S_j$), is a solution. Berg and Opfer (2013) extended this framework to negative start values and proved (their Theorem 2.3) that the Collatz conjecture for $s$ — every $n\ge 1$ has some iterate in $S_1\cup S_2\cup S_3$ — is equivalent to the statement that there are exactly three linearly independent holomorphic solutions with coefficients in $\{0,1\}$. If the conjecture fails, some start value belongs to none of the three basins, its trajectory either cycles in a new cycle or diverges to infinity, and the indicator of its merge class is a fourth linearly independent solution. The paper also proposed a descent reformulation (its Theorem 3.8): the conjecture follows if every so-called $2\ell$ table — the subsequence of even entries of the trajectory of an even start value $2\ell\ge 4$ — contains an entry smaller than $2\ell$ itself. This is a one-way sufficient condition, not an equivalence, and as literally stated the descent property does not hold for even start values lying inside the known cycles (e.g. $10$ and $34$), whose even iterates never drop below the start value. Finally, the paper conjectured (its Conjecture 2.4) that $\eta_1,\eta_2,\eta_3$ have the unit circle $\{|z|=1\}$ as a natural boundary, meaning they admit no analytic continuation across any point of the circle. This companion conjecture is now a theorem: Bell and Lagarias (2015) identified $\eta_1,\eta_2,\eta_3$ with backward-orbit generating functions $f_{-1,1},f_{-1,5},f_{-1,17}$ of the $3x-1$ map and proved that $f_{-1,m}$ has the unit circle as a natural boundary for every $m\ge 1$, using the Pólya–Carlson dichotomy (a power series with integer coefficients and radius of convergence $1$ is either rational or has the unit circle as natural boundary) together with the Skolem–Mahler–Lech theorem. What their method leaves untouched, and what remains open, is the three-dimensionality of the solution space, i.e. the $3x-1$ conjecture itself.

Problem Statement

Let $s:\mathbb{N}\to\mathbb{N}$ be the Collatz function for negative start values, written on positive integers: $s(n)=n/2$ if $n$ is even and $s(n)=(3n-1)/2$ if $n$ is odd, with iterates $s^{(k)}$. Let $S_1=\{1\}$, $S_2=\{5,7,10\}$, and $S_3=\{17,25,37,55,82,41,61,91,136,68,34\}$ be the three known cycles. The $3x-1$ conjecture asserts: for every $n\in\mathbb{N}$ there exists $k\ge 0$ such that $s^{(k)}(n)\in S_1\cup S_2\cup S_3$. Equivalently (Berg–Opfer, Theorem 2.3): the linear space spanned by holomorphic functions $h(z)=\sum_{n\ge 1}h_n z^n$ on the open unit disk with Taylor coefficients $h_n\in\{0,1\}$ satisfying $h_n=h_{s(n)}$ for all $n\ge 1$ — equivalently, satisfying the functional equations

$$h(z)+h(-z)=2h(z^2),\qquad h(z^3)-h(-z^3)=\frac{2z}{3}\sum_{\nu=0}^{2}\lambda^{-\nu}h(\lambda^{\nu}z^2),\ \ \lambda=e^{2\pi i/3}.$$

has dimension exactly three (a basis being the three basin generating functions $\eta_1,\eta_2,\eta_3$). Resolve this: either prove that the dimension is exactly three (equivalently, that every positive integer's trajectory enters one of the three known cycles), or refute it by exhibiting a fourth linearly independent solution — equivalently, a positive integer whose trajectory under $s$ enters a cycle other than $S_1,S_2,S_3$, or a positive integer whose trajectory diverges to infinity. The companion question whether $\eta_1,\eta_2,\eta_3$ have the unit circle as a natural boundary is already settled affirmatively (Bell–Lagarias 2015) and is not part of what remains to be solved.

References

1. Lothar Berg and Gerhard Opfer, "An Analytic Approach to the Collatz $3n+1$ Problem for Negative Start Values", Computational Methods and Function Theory 13 (2013), no. 2, 225–236. DOI: 10.1007/s40315-013-0017-z. https://doi.org/10.1007/s40315-013-0017-z
2. Lothar Berg and Gerhard Opfer, "An analytic approach to the Collatz $3n+1$ problem for negative start values with an appendix of tables", Hamburger Beiträge zur Angewandten Mathematik, Preprint Nr. 2012-11, Universität Hamburg, December 2012. https://preprint.math.uni-hamburg.de/public/papers/hbam/hbam2012-11.pdf (extended preprint of the CMFT paper)
3. Jason P. Bell and Jeffrey C. Lagarias, "3x+1 inverse orbit generating functions almost always have natural boundaries", Acta Arithmetica 170 (2015), no. 2, 101–120. DOI: 10.4064/aa170-2-1; arXiv:1408.6884. https://doi.org/10.4064/aa170-2-1
4. Lothar Berg and Günter Meinardus, "Functional Equations Connected With The Collatz Problem", Results in Mathematics 25 (1994), 1–12. DOI: 10.1007/BF03323136. https://doi.org/10.1007/BF03323136
5. Lothar Berg and Günter Meinardus, "The $3n+1$ Collatz problem and functional equations", Rostocker Mathematisches Kolloquium 48 (1995), 11–18. Zbl 0861.11008. https://zbmath.org/?q=an%3A0861.11008
6. John H. Conway, "On Unsettleable Arithmetical Problems", The American Mathematical Monthly 120 (2013), no. 3, 192–198. DOI: 10.4169/amer.math.monthly.120.03.192. https://doi.org/10.4169/amer.math.monthly.120.03.192
7. Christos N. Efrem, "Functional Equations for Generalized Collatz Dynamics Using Integral Representations and Residue Calculus", arXiv:2510.06736 (2025). https://arxiv.org/abs/2510.06736