QUADRATIC EXCLUSION LAWS FOR CONSECUTIVE ARTIN PRIMES IN ARBITRARY BASES

Abstract. Let a ≥ 2 be a non-square integer with squarefree part d, and say that an oddprime p is Artin base a if a is a primitive root modulo p. (We restrict to odd primes throughout:modulo 2 every odd integer trivially has full multiplicative order, and the quadratic obstructionbelow fails there.) Membership in Aa forces the quadratic character value χa(p) = −1, andsince χa has conductor f, this necessary condition depends only on p mod f. We determine,for every non-square base a, the exact set of gap classes on which the resulting quadraticobstruction forbids two primes from both being Artin base a. The forbidden classes themselvesare not new: Tinkov´a, Waxman and Zindulka [15], studying Artin prime pairs at a fixed evenshift, gave sufficient conditions on (d, g) for the pair set to be empty, and we have verified thattheir conditions and our classification agree on every non-square base a ≤ 80, the exceptionalsquarefree part d = 5 included. Our contribution to the classification is therefore one ofmethod and of completeness rather than of discovery: we derive the classes from an exactcounting identity, obtain the converse direction — that outside the listed cases no quadraticexclusion class exists — as a theorem rather than a conjecture, cover bases excluded fromtheir hypothesis g ∈ H1 (perfect powers such as a = 8, 27, 32), and machine-check the keysteps in Lean 4. Two mechanisms produce such exclusion classes. The first is reversal:χa(q) = −χa(p) for all admissible p, q = p + g, so one of the two primes is a quadratic residuebase a; we classify the reversing classes completely via the factorization of the discriminantof Q(√a) into prime discriminants. The second, operating when no reversing class exists,is inadmissibility. Its analysis rests on an exact count, and we prove the general identity4N−−(g) = T(g) − A(g) − B(g) + S(g) valid for every modulus, prime or composite, fromwhich the prime-conductor evaluation N−−(g) = (d − 3 + 2χ(g))/4 for g ̸≡ 0 (mod d) (andN−−(0) = (d − 1)/2) follows as a special case. The resulting complete dichotomy is: basea admits a quadratic exclusion class if and only if d is even, d ≡ 3 (mod 4), 3 | d, or d = 5.We emphasise throughout that the quadratic obstruction is necessary but not sufficient forArtin status, so an exclusion class is a proof of impossibility, while the absence of one isnot a proof of possibility. The classification is verified by exhaustive computation for allnon-square 2 ≤ a ≤ 80, and the exclusion laws hold without exception across 84,981,870base–pair incidences below 109. The counting identity, together with the d = 5 and d = 13cases of the prime-conductor count, is additionally machine-checked in Lean 4.We then measure, for eleven bases, the consecutive-pair Artin correlation δ(a) over all50,847,531 consecutive prime pairs below 109 with smaller member at least 5. The exclusionweight w correlates with |δ| at r = 0.646, but this association is confounded with the conductor:r(log f, |δ|) = −0.957, the partial correlation of w given log f is only 0.136, and the three baseswith no exclusion classes still carry |δ| up to 0.043. Restricting δ to gap classes outside theexclusion set yields a positive value for all eight bases that possess such classes. Among thequantities compared, the conductor of χa is the stronger descriptive predictor of the globalcorrelation; we do not claim it is the operative mechanism.