68 problems

Collatz conjecture. For every $x \in \mathbb{N}$ there exists $N \ge 0$ such that $\phi^{N}(x) = 1$. This is the classical Collatz problem, asserting that every positive integer ev…

For $m\in\mathbb{Z}^+$, let $\tau(m)$ be the minimal $n\in\mathbb{N}$ such that $\mathsf{T}^n(m)=1$, when such an $n$ exists, and set $\tau(m)=\infty$ otherwise. Here $\log_2$ deno…

Let $A\in\mathbb{F}_2[x]\setminus\{0\}$. Define the associated sequences $(a_{2k})_k$, $(A_{2k})_k$, and $(A_{2k+1})_k$ by $A_0=A$, $A_{2k}=1+(x^2+x+1)A_{2k-1}$, and by removing al…

Let $\mathbb{Q}_{odd}$ be the ring of rational numbers whose reduced denominators are odd, identified with the subring $\mathbb{Q}_2\subset\mathbb{Z}_2$ of rational 2-adic integers…

$(3x+1)$-conjecture. For every $x\in\Pi$, there is an integer $n$ such that

$(3x+1)$-Conjecture. For every $x\in\Pi$, there is an integer $n$ such that

The 3x+1 conjecture. The basin of attraction of this cycle is the entire set of positive integers.

Let $M_2(\mathbb{C})$ be the observable algebra of the one-qubit alphabet, with Pauli matrices $\vec{\sigma}=(\sigma_1,\sigma_2,\sigma_3)$ and identity matrix $\mathbb{I}$. For…

For positive integers $n$, define … Let $t(n)$ denote the relevant quantity associated with the iteration of $T'$ as in the source. The independent-random-walk growth conjecture. ……

The $3n+1$ conjecture. Starting with any $n\in\mathbb{N}$, there is a positive integer $k\in\mathbb{N}_0$ such that $C^{(k)}(n)=1$.

Consider the map $H:^+\to^+$ defined by … Starting from $x=8$, classify each iterate as odd or even. Antihydra's odd-even frequency conjecture. At no point in the iteration are the…

Let $\mathbb{Z}_2$ denote the $2$-adic integers, let $F_2[[q]]$ be the ring of formal power series over $F_2$, and let $T$ and $T_{1,1+q^2}$ be the maps in the source. A conjugacy…

Let $\mathbb{Z}_2$ denote the $2$-adic integers, let $F_2[[q]]$ be the ring of formal power series over $F_2$, and let $T$ and $T_q$ be the maps in the source. A conjugacy is a bij…

Let $T$ act on the positive integers by … The $T$-orbit of every positive integer contains $1$. The $3x+1$ conjecture. Every positive integer eventually reaches $1$ under iteration…

Nilpotency conjecture for the Collatz adjacency submatrices. If $n\geqslant 3$, then the matrix $C_n$ is nilpotent.

For $m\in\mathbb{Z}^+$, let $\tau(m)$ be the minimal $n\in\mathbb{N}$ such that $\mathsf{T}^n(m)=1$, when such an $n$ exists, and set $\tau(m)=\infty$ otherwise. Kontorovich–Lagari…

Let $T(n)$ denote the paper's count of Collatz integers in the interval $[2^n+1,2^{n+1}]$, and let $\delta(n)$ be the formula defined in the preceding discussion for the number of…

Let $C_n(L)$ denote the Collatz chain associated with an odd integer $L$ in the interval under discussion, and let $\alpha(C_n)$ and $\beta(C_n)$ denote the numbers of applications…

Triangle Conjecture. There exists no $a\in T^{-k}(3^kh-1)$ such that $a\neq2^kh-1$ and

Let $A=x^n+x+1$ with $n\geq 2$, and let $\ell_A$ denote the length of its binary-polynomial Collatz transformation sequence. Let $s$ be the greatest integer such that…

Let $A=x^n+x+1$ with $n\geq 2$, and let $A_1,A_3,\ldots$ be the odd subsequence in the binary-polynomial Collatz transformation. Let $s$ be the greatest integer such that…

Let $M=x^2+x+1$ and let $\ell_A$ denote the length of the Collatz transformation sequence associated with a nonzero binary polynomial $A\in\mathbb{F}_2[x]$. Power-family length con…

Let $x$ be a positive integer. Consider the Collatz process, which sends an even integer to $x/2$ and an odd integer to $(3x+1)/2$. A computation sequence is reduced dynamics when…

A finite holdout problem specifies a finite set of primes to be retained after applying a linear function, and a starting value determines the resulting sequence. Finite holdout no…

Let a Collatz permutation sequence be a trajectory under one of the permutation functions considered in the paper, and let the four known cycles be the cycles identified for these…