11 problems

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…

Let $m=8+12q$ with $q=61+64d$ for some nonnegative integer $d$. Define the sequence $\{g_l\}_{l\ge 10}$ as in the preceding definition, with $g_{10}$ and $g_{10}'$ selected accordi…

Let $M$ be the map used to embed $\mathbb Z_2$ into $[0,2]$, and let $Q$ be the 3x+1 conjugacy map. Define the 3x+1 set by … A point of this set has rational coordinates when both…

Let $Q$ be the 3x+1 conjugacy map on $\mathbb Z_2$, and call a periodic orbit an odd cycle when all its elements are odd. Odd Cycles Conjecture. The function $Q$ has exactly two od…

Let $Q$ be the 3x+1 conjugacy map, and call a periodic orbit of $Q$ an odd cycle when all its elements are odd 2-adic integers. 3x+1 Conjugacy Finiteness Conjecture. For every inte…

Let $Q$ be the 3x+1 conjugacy map on $\mathbb Z_2$. Call a fixed point odd if it is an odd 2-adic integer. Bernstein's Fixed Point Conjecture. The only odd fixed points of $Q$ are…

Let $Q$ be the 3x+1 conjugacy map on $\mathbb Z_2$, and let the ergodic domain of $Q$ be the union of the invariant sets on which $Q$ is ergodic. Ergodicity Conjecture. The ergodic…

Let $\mathbb Z^+$ denote the positive integers, let $\mathbb Z_2$ be the 2-adic integers, and let $Q$ be the 3x+1 conjugacy map. The 3x+1 problem. One should have … or equivalently…

Let $\mathbb Z_2$ denote the ring of 2-adic integers, and let $Q:\mathbb Z_2\to\mathbb Z_2$ be the 3x+1 conjugacy map. A 2-adic integer is rational when it belongs to $\mathbb Q$.…

Let $\hat{\Omega}_{m,r}(x)$ be the real limit, when it exists, of the truncated generalized autoconjugacies $\Omega_{k,m,r}(x)$, and let $\Omega_{m,r}$ be the corresponding $2$-adi…

Let $m$ and $r$ define the generalized Collatz map $T_{m,r}$, with associated maps $Q_{m,r}$ and $\Omega_{m,r}$, and let $\mathbb{Q}_2$ denote the field of rational $2$-adic number…