It's that time of year again when I manage to convince myself I've cracked the notorious Collatz conjecture... Only this time I really have! So let $I(i,j)$ be the intersection of the balls $B_{2^{-i}}(-1)$ and $B_{2^{-j}}(0)$ of natural numbers in the 2-adic metric. For any $i,j$ then $I(i-1,j)$...

I asked about the very similar $\pi(ab)\geq \pi(a)\pi(b)$ a while ago, and this is indeed a proven result for $a,b\geq \sqrt{53}$. Empirically, the stricter $\pi(ab)>\pi(a)\pi(b)$ looks true so long as either $a$ or $b$ is $\geq 23$. It also seems to pretty immediately fall out of the PNT, a la $...