Twin Gauss Circle Primes

Prove that there exist infinitely many integers r ≥ 1 such that both C(r) and C(r + 1), where C(r) is the number of integer lattice points inside the radius‑r circle centered at the origin, are prime numbers.

Background

The paper defines Gauss Circle Primes as those values of C(r) that are prime. Drawing an analogy to the classical twin prime conjecture (which asserts infinitely many prime pairs p and p+2), the authors propose a geometric analogue based on consecutive radii r and r+1.