Under Goldbach's conjecture, denote by $r_{0}(n)$ the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime. Say two composite integers $a$ and $b$ are primally entangled if $r_{0}(a)\mid b$ and $r_{0}(b)\mid a$.
So let $a$ and $b$ be two primally entangled integers and consider now the following two sequences: $(u_{n})_{n\geq 0}: u_{n+1}(a,b):=r_{0}(u_{n}(a,b)+v_{n}(a,b))$ with $u_{0}(a,b):=a+b$ $(v_{n})_{n\geq 0}: v_{n+1}(a,b):=r_{0}(u_{n}(a,b).v_{n}(a,b))$ with $v_{0}(a,b):=ab$.
Is there an integer $N=N(a,b)$ such that $\{u_{N}(a,b),v_{N}(a,b)\}=\{0,1\}$? This would be a kind of convergence in finite time towards the neutral elements of the rig (ring without negatives) $(\mathbb{N},+,\times,0,1)$.