Conjecture:
For all primes of the form $a^2+b^2$, there are natural numbers $s,t,u,v$ such that
- $\quad s^2+t^2,u^2+v^2$ are primes
- $\quad a+b=s+t$
- $\quad u+v=s+t+2$
- $\quad |s-u|+|t-v|=2$
This conjecture implies Any odd number is of form $a+b$ where $a^2+b^2$ is prime and seems to be much stronger. I have tested it for $a+b<50.000$ without founding a counterexample, but I guess there may exist counterexamples.
In the diagram the vertices correspond to points $(x,y)$ where $x^2+y^2$ is prime, the red edges correspond to pairs of vertices with taxi cab distance equal to $2$ and the gray diagonal lines is the (taxi cab) equidistance lines to origin.
The conjecture says there are paths following red edges and gray diagonal lines up to any distance from origin.
Addendum: Further tests show that there exists red vertical edges that together with the gray diagonal lines make such paths for all odd $n=a+b=3,\dots,49,999$ except for $n\in\{9,19,49,75,149\}$.
In this case I want help with finding counterexamples.