9 problems

The 3x+1 conjecture. Every integer $n\geq 1$ eventually reaches $1$ under iteration of $T$.

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 $\mathbb Z^+$ 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 $\Phi$ be the continuous conjugacy function on the $2$-adic integers associated with the $3x+1$ map. A point is periodic of period $m$ when its least positive period under…

Let $T:\mathbb{Z}_{2}\to\mathbb{Z}_{2}$ be the $3x+1$ map, and let $\Phi$ be the continuous function conjugating $T$ to the $2$-adic shift map. An element of $\mathbb{Z}_{2}$ is od…