97 problems

Let $C:\mathbb{Z}\to\mathbb{Z}$ be the Collatz map, and let $\mathbb{N}_{1}$ denote the positive integers. A periodic point is an integer lying on a finite cycle of $C$. Weak Colla…

Stopping-time equality conjecture. If that bound condition on $d$ is unnecessary or automatically satisfied, then

The $3x+1$ conjecture. Every positive integer has the tri-convergence property. This is presented as a reformulation for the complex Collatz extension, not as a resolution of the o…

Autoconjugacy Conjecture.

Weak $3x+1$ conjecture. The semigroup ${\cal R}$ contains every positive integer. This is a weak version of the $3x+1$ problem. The bibliography attributes the formulation to Farka…

Rationality conjecture for $Q_a$. The map $Q_a$ maps the rationals with odd denominators to rational numbers. The bibliography states that this conjecture is true for $a=\pm1$, fal…

The $3x+1$ conjecture. Every integer $m\geq1$ has some iterate $T^{(k)}(m)=1$. This is the central unsolved problem discussed in the bibliography.

Let $T:\mathbb Z\to\mathbb Z$ be given by … An orbit of an integer $n$ is the sequence $n,T(n),T^{(2)}(n),\ldots$. The bounded-orbit conjecture. All orbits of the $3x+1$ iteration…

Let $T:\mathbb Z\to\mathbb Z$ be given by … For a positive integer $n$, write $T^{(k)}(n)$ for its $k$-th iterate. The $3x+1$ conjecture. For every positive integer $n$, there exis…

Let ${\mathbb R_1}$ be the positive real numbers, let ${\mathbb N_1}$ be the positive integers, and let $U:{\mathbb R_1}\to{\mathbb R_1}$ be the paper's real 3x+1 map. Define … and…

Let ${\mathbb R_1}$ denote the positive real numbers, and let $F:{\mathbb R_1}\to{\mathbb R_1}$ be the piecewise extension of the Collatz function defined by … Write…

Let $T:\mathbf{Z}\to\mathbf{Z}$ be the $3x+1$ function defined by … For $n\geq1$, write $T^{(k)}(n)$ for the $k$-fold iterate of $T$. The 3x+1 conjecture. For each $n\geq1$, some i…

Let the fiber-$57$ return dynamics be the subsequence of the odd-to-odd Syracuse dynamics consisting of returns to the fiber described in the paper, and let the sustaining core be…

Let $M$ be the modular depth, and let $(n,V)$ be a net-positive parameter pair for a potential cycle, where $n$ is the cycle length and $V$ is the total valuation contribution. The…

Let $G_M^+$ denote the augmented return graph at modular depth $M$, incorporating the mixed-state excursion, exit, and return dynamics, and let $K\ge1$ be the parameter indexing th…

Let $(n_0,n_1,\ldots)$ be a Collatz orbit with $n_0>1$. Let $\tau_1<\tau_2<\cdots$ be successive fiber-$57$ return times, set $q_j=\lfloor n_{\tau_j}/64\rfloor$, and let $I_r$ be t…

Let $\Delta_i$ be the odd-skeleton drift increments along a Collatz orbit, and let $\operatorname{Cov}_{\mathrm{orbit}}(\Delta_0,\Delta_h)$ denote their orbit covariance at lag $h$…

Let $\mu_K$ be the orbit measure on $\mathbb{Z}/2^K\mathbb{Z}$, let $\operatorname{hw}(\xi)$ denote Hamming weight, and define the spectral content at weight $w$ by … Spectral diff…

Let $\eta_K^{\mathrm{renew}}(n_0)$ denote the total discrepancy contribution from renewed depth-$K$ alignment events for an orbit of the odd-to-odd Syracuse map starting at odd…

Let $\eta_K(n_0)$ be the phantom gain discrepancy at depth $K$ for the odd-to-odd Syracuse orbit starting at odd $n_0$. Gain-observable recurrence conjecture. There exist constants…

Let $\delta_K(n_0)$ denote the total-variation discrepancy at modular depth $K$ for the orbit starting at odd $n_0$. Depthwise discrepancy recurrence conjecture. There exist consta…

Let $T$ be the odd-to-odd Syracuse map, let $n_0$ be odd, and let $h_K$ be the phantom gain observable at depth $K$, with reference mean $R(K)$. Define … Observable-specific weak-m…

Let $T$ denote the odd-to-odd Syracuse map, let $n_0$ be odd, and define $a_i=T^i(n_0)\bmod 2^M$. Let $\mu_{\mathrm{orb},N}=\frac{1}{N}\sum_{i=1}^N\delta_{a_i}$ and let $\mu_U$ be…

For $p\geq0$, let $(d_p,\alpha_p,\beta_p)_+=(2^{p+1},3\cdot2^p,2^p)_+$ and let $T_p$ be its associated mapping. Diagonal generalized Collatz conjecture. If $p\geq0$ and $p\neq2$, t…

For integers $p\geq q\geq0$, set … and let $T_{p,q}$ be the associated mapping. A triplet is strongly admissible when it has finitely many cycles and no divergent trajectory. Gener…