Infinitely many prime pairs (p, q) with q = p^2 − p + 1

Determine whether there exist infinitely many pairs of distinct primes (p, q) satisfying q = p^2 − p + 1, which, by Proposition 2.4, would yield infinitely many non-abelian groups of order pq (specifically semidirect products C_p ⋊ C_q) with fcod(G) = 1.

Background

The paper studies the sum of character codegrees Sc(G) and the normalized invariant fcod(G) = Sc(G)/|G|. In Section 2 the authors focus on the special case fcod(G) = 1 and analyze the structure of groups satisfying this condition.

Proposition 2.4 shows that for a non-abelian group of order pq (with p, q distinct primes), namely a semidirect product C_p ⋊ C_q, one has fcod(G) = 1 if and only if q = p^2 − p + 1. The authors list examples of such prime pairs (2,3), (3,7), and (13,157), but explicitly state uncertainty about whether there are infinitely many such pairs, a number-theoretic question that impacts the classification of groups with fcod(G) = 1 in this family.