CORRELATIONS BETWEEN PRIMITIVE ROOT STATUSES OF CONSECUTIVE PRIMES

Abstract

Abstract. Call a prime p an Artin prime for base 10 if 10 is a primitive root modulo p. Weinvestigate the dependence between the Artin statuses of consecutive primes. Computing Artinstatus for all 50,847,531 primes 7 ≤ p ≤ 109, we find that consecutive primes are measurablyanticorrelated: P(Artn+1 | Artn) = 0.36514 versus P(Artn+1 | ¬Artn) = 0.37928, a deficit δ =−0.01414 (Pearson statistic 10,167 on one degree of freedom; signed square root −100.8 under anominal independent-status reference model; over 50,847,530 consecutive pairs). The dependencehas striking gap structure. We prove an elementary exclusion law: if p,p + g > 5 are both primewith g ≡ 20 (mod 40), then 10 is a quadratic residue modulo exactly one of them, so they cannever both be Artin primes for base 10; empirically, among 2,195,882 consecutive pairs with gap20 or 60 there is indeed not a single doubly-Artin pair; an independent recomputation extends thisto 194,296,748 such pairs below 1011, again with none. Conversely, gaps g ≡ 0 (mod 40) forceequal quadratic-residue status and exhibit strong positive dependence (P(Artn+1 | Artn) = 0.899at g = 40). Conditioning on the joint residues of (pn,pn+1) modulo 120 or 840 substantially reducesa selected-cell signed residual association; the remaining association is small in that metric but isnot established to vanish. These results are consistent with a substantial role for arithmetic residuestructure in the observed anticorrelation; they do not constitute a complete decomposition or anindependent quantitative verification of the Hardy–Littlewood explanation. As an intermediatelink we report the (apparently unreported) anticorrelation rω(pn −1),ω(pn+1 −1) = −0.041. Atfour decade-spaced cutoffs x = 108,109,1010,1011 the deficit grows in magnitude with decreasingincrements, δ = −0.01336,−0.01414,−0.01472,−0.01500, and no turnover is observed; we drawno asymptotic conclusion from this, and in particular these finite-range values do not distinguisheventual decay to zero from persistence of a nonzero correlation. We formulate open questions anddiscuss extensions to general bases.