# Primality radii in arithmetic progression

Under Goldbach conjecture, say a positive integer $$r$$ is a primality radius of a large enough composite integer $$n$$ if and only if both $$n-r$$ and $$n+r$$ are prime. Let for given $$n$$ the quantity $$N_{2}$$ to be the number of primality radii of $$n$$ and $$k$$ the greatest positive integer not exceeding $$\sqrt{N_2}$$ (the assumption of GC entails this number always exists).

Can one always find a $$k$$-term sequence of primality radii of $$n$$ in arithmetic progression with minimal positive common difference ?

This is not true, even without the minimal positive common difference condition.

The first counterexample is $$n=81$$, which has $$10$$ primality radii that contain no arithmetic progression of length $$3$$: $$\{2,8,20,22,28,50,58,68,70,76\}$$

More generally, we can note that if the radii are in AP, then so are the equivalent $$n-r$$ values. That is, we are also finding primes in AP. The number of radii grows relatively quickly and the required $$k$$ soon exceeds the longest known prime APs. The current record prime AP has length $$26$$, but:

$$n=12285, N_2=690, k=26$$
$$n=13650, N_2=738, k=27$$