32 problems

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

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…

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…

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

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…

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 $\operatorname{sft}(n)$ denote the Syracuse falling time of an odd positive integer $n$. Large-number Syracuse falling-time conjecture. … for all odd $n\geq 2^{5000}$. This con…

Let $\operatorname{ft}(n)$ denote the falling time associated with the accelerated $3x+1$ map. Large-number falling-time conjecture. … for all $n\geq 2^{500}$. This is a strong qua…

Let $\operatorname{sft}(n)$ denote the Syracuse falling time of an odd positive integer $n$. Mersenne Syracuse falling-time conjecture. … for all $\ell\geq 4625$. The assertion agr…

Let $\operatorname{syr}$ be the Syracuse map on odd positive integers, and let $\operatorname{sft}(n)$ denote the Syracuse falling time of an odd integer $n\geq 3$, namely the leas…

Let $n$ be a positive integer, and consider the Collatz dynamic on odd positive integers. The Golden Automaton is the finite proof system described in the source for establishing C…

Let … Let ${\cal O}_1=\{1,2\}$. An eventual finite orbit is a finite orbit in which the iterates of an initial integer eventually remain. Collatz orbit conjecture. ${\cal O}_1$ is…

Let $f$ be the Collatz function and let $\{f^{s}(a)\}_{s=1}^{\infty}$ be the corresponding Collatz process. Let $\Omega(a)$ denote the total number of prime factors of $a$, counted…

Let $f$ be the Collatz function and let $\{f^{s}(b)\}_{s=1}^{\infty}$ be a full Collatz process. The odd-prime occurrence conjecture. There exists an odd prime $p$ such that … The…

Let $f$ be the Collatz function, let $p>2$ be prime with $p\equiv 3\pmod 4$, and let $\mu$ denote the Möbius function. The Möbius-nonvanishing conjecture. There exists an integer…

Let $f$ be a Collatz function and let $\{f^{s}(b)\}_{s=1}^{\infty}$ be the corresponding Collatz process. A process is full in the sense used by the source, and define … Let $\rho$…

Let $f$ be the Collatz function and let $\{f^{s}(b)\}_{s=1}^{\infty}$ be a full Collatz process. The prime-generator index–order conjecture. If the process converges, then … Here…

Let $T_R:\mathbb{Z}\to\mathbb{Z}$ be defined by … For $k\geq 0$, define $T_R^{0}(n)=n$ and $T_R^{k+1}(n)=T_R(T_R^{k}(n))$, and call $(T_R^{k}(n))_{k\geq 0}$ the $T_R$-trajectory of…

Let $(a_n)_{n\geqslant 1}$ be an E-sequence, meaning an infinite sequence of positive integers, and define … An E-sequence is Ω-convergent to an odd positive integer $x$ if it is t…

For any odd positive integer $x_0$, define its trajectory $(x_n)_{n\geqslant 0}$ by choosing positive integers $a_n$ such that … with every $x_n$ odd. A trajectory is pure periodic…

Rationality criterion. The power series $G_\psi(T)$ is rational if and only if $G_\psi(\frac{1}{3})$ is a rational element of ${\bf Q}_2$. This is presented as a reformulation of t…