50 problems

Generalized Collatz convergence and cycle conjecture. For all $p\geq0$ with $p\ne2$, the triplet $(2^{p+1},3\times2^p,2^p)_+$ is strongly admissible of order one, with unique trivi…

Mersenne-triplet admissibility conjecture. For all $p\geq 2$, $(2^{p-1},2^{p}-1,1)_{\pmb{+}}$ is an admissible triplet of order one. It is strongly admissible only for $p=2$ and we…

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 a positive integer $n\geq1$, if $n$ is a multiple of $10$, replace it by $n/10$; otherwise, let $r$ be its last digit, replace it by $(12n+8r)/10$, and iterate this process. Mo…

Let $\operatorname{sat}(k)$ be the sequence defined in the paper, and let $\langle\operatorname{sat}(k)\rangle$ denote its associated word over the alphabet $\{1,2\}$. The digit-fr…

Let $T:\mathbb N\to\mathbb N$ be the Collatz function, let $T^{(k)}$ denote its $k$-fold iterate, and fix $n_0\in\mathbb N$ with $n_0\geq 1$. The even-iterate frequency conjecture.…

Let $d_{\text{T}}(n)$ denote the delay and $M_{\text{T}}(n)$ the maximum excursion of the Collatz sequence starting at the positive integer $n$. Delay and maximum-excursion bound c…

Let $T$ be the Collatz map. A sequence $n,T(n),\ldots,T^j(n)$ is paradoxical when $C_j(n)<1$ and $T^j(n)\geq n$. Nonexistence conjecture for large paradoxical sequences. There is n…

Let $q$ be an odd prime, and let $\chi_q:\mathbb{Z}_2\to\mathbb{Z}$ denote the numen of the shortened $qx+1$ map. Divergence conjecture for $qx+1$. There exists a…

Let $d\geq2$, let $m_{0},\ldots,m_{d-1}$ be nonzero integers, and choose $r_j\in\mathbb{Z}$ with $r_j\equiv jm_j\pmod d$. Define $T_{\mathrm{Matthews}}:\mathbb{Z}\to\mathbb{Z}$ by…

Fix coprime integers $m,d>1$ with $m>d$, and define $T_{\mathrm{Möller}}:\mathbb{Z}\to\mathbb{Z}$ by the residue-class formula given in the source. Möller's conjecture. The map…

Let $C:\mathbb{Z}\to\mathbb{Z}$ be the Collatz map and let $\mathbb{N}_{1}$ denote the positive integers. A trajectory is divergent when its iterates are unbounded. Divergent traje…

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…

For each $x\in(0,\infty)$, let $G_x$ be the directed graph constructed from the balanced ternary data associated with $x$. A branch is a directed path in $G_x$ of the type used in…

Let $C$ denote the real Collatz map used in the paper, and let $x\in(3/4,\infty)$. A subset $A$ of a measurable set $B\subseteq\mathbb{R}$ is Lebesgue-co-null in $B$ when…

Let $\operatorname{Col}_R$ denote the real analogue of the Collatz map considered in the paper, and let $r$ be a real number satisfying $r>9/4$. Bounded-interval conjecture. There…