We prove a quadratic dichotomy law for the prime factors of Nₐ (q) = (aq+1) / (a+1) = Φ₂q (a), where a and q are odd primes. If a prime p not dividing 2a (a²−1) divides Nₐ (q), then ordₚ (a) = 2q and, writing p = 2qm+1, the Legendre symbol (p/a) equals +1 unconditionally when a ≡ 3 (mod 4), while for a ≡ 1 (mod 4) it equals +1 exactly when m is even, equivalently when p ≡ 1 (mod 4). The proof uses only Euler's criterion and quadratic reciprocity. The case a = 3 shows that every prime factor p > 3 of F (n) = (3^ (2n+1) +1) /4 satisfies p ≡ 1 (mod 3) ; consequently a cubic refinement via reciprocity in Zω applies to all such factors without restriction, sharpening the Birkhoff–Vandiver congruence from modulo 2q to modulo 6q. A single underlying mechanism (an ℓ-th power level lemma) governs both the quadratic and cubic cases, and explains, via the finiteness of the unit group of Zζ_ℓ for ℓ ∈ 2, 3 only, why no elementary congruence analogue is available in higher degree. Motivated by the Sophie Germain pairing of the Mersenne number M (2n+1) = 2^ (2n+1) −1 with F (n), we justify the heuristic independence of the two primality events by the linear disjointness of the Kummer extensions Q (ζq, 2^ (1/q) ) and Q (ζ₆q, 3^ (1/2q) ), and derive an expected count below 2×10⁻² of simultaneously prime pairs with n > 3, consistent with the known instances n = 2, 3. All statements are verified computationally: the dichotomy law on 337 factors across thirteen bases, and the cubic refinement on all 65 common prime divisors of M (2n+1) and F (n) for Sophie Germain primes n ≤ 2×10⁶. This work develops and generalizes the cyclotomic results first presented in DOI 10. 5281/zenodo. 20624434.
Hamza BESSAOUDI (Mon,) studied this question.