Primitive roots and controlled prime predecessors

Source and verification

This article explains family 029 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Multiplicative order and admissible bases

For a prime $p\nmid a$, the order ${\mathrm{ord}}_p(a)$ is the least positive $k$ such that $a^k\equiv 1(\mathrm{mod}p)$. Fermat’s theorem implies that it divides $p-1$. The base is a primitive root when its order equals $p-1$.

For instance, the powers of two modulo seven are $2,4,1$, so its order is three. Modulo five they are $2,4,3,1$, so two is a primitive root.

An integer $a$ is admissible if it is not $-1$ and is not an integer square. Squares have order dividing $(p-1)/2$ at odd primes not dividing them, while $-1$ has order at most two. These are necessary obstructions to infinitude.

Claimed one-base result

For every fixed admissible integer $a$, the first manuscript asserts $c_a>0$ such that, in every sufficiently large interval $(x,2x)$,

$$\begin{array}{c}\#\left\{p:x<p<2x,\begin{array}{c}p\nmid a\\ {\mathrm{ord}}_p(a)=p-1\end{array}\right\}\\ \ge c_a\frac{x}{(\log x{)}^2}.\end{array}$$

This proves infinitude within the manuscript’s claimed theorem. It does not give Artin’s predicted positive-density asymptotic of order $x/\log x$. Constants and thresholds may depend on $a$. (OpenAI, 2026)

🏷️ Detecting an order defect

Define the index

$$I_p(a)=\frac{p-1}{{\mathrm{ord}}_p(a)}.$$

For a prime $q\mid p-1$,

$$\begin{array}{c}q\mid I_p(a)\\ ⟺\\ a^{(p-1)/q}\equiv 1(\mathrm{mod}p).\end{array}$$

Indeed, the displayed congruence is equivalent to the order dividing $(p-1)/q$. Thus proving maximal order amounts to excluding every prime divisor of $I_p(a)$.

A quadratic-nonresidue condition excludes $q=2$. For the other $q$‘s, the congruence has an interpretation as complete splitting in a field obtained by adjoining $q$th roots of unity and a $q$th root of $a$. The source uses this interpretation to estimate the exceptional primes. These notes need only the congruence criterion, not the algebraic theory of the splitting fields.

🏷️ Two inputs that must meet

One analytic ingredient is a separate zero-free theorem claimed in the first manuscript: for every cyclotomic field $F$ containing the twelfth roots of unity, and every finite-order Hecke character $\eta$,

$$\begin{array}{c}{L}_F(s,\eta )\ne 0\\ (\mathrm{Re}s>1-10^{-6}).\end{array}$$

Principal poles at one are allowed. The numerical width is common to the stated fields and characters, while intermediate constants and thresholds may depend on fixed $F,\eta$.

This is a different family of $L$-functions from the Dirichlet and quadratic-field results in family 003. The manuscript develops cubic-theta coefficients and reciprocal-$L$ detection over each required field; it does not obtain the assertion merely by citing the Dirichlet result.

The second ingredient constructs many primes with controlled predecessors:

$$\begin{array}{c}p=crQ+1,\\ c\in \{2,4\},\\ Q>x^{0.9}\end{array}$$

with $Q$ prime and every prime factor of $r$ between $\exp ((\log x{)}^{0.1})$ and $\exp ((\log x{)}^{0.3})$. A fixed progression ensures that $a$ is a quadratic nonresidue.

Any remaining order defect has a prime divisor among those of $rQ$. The large factor $Q$ is handled by an elementary exceptional-set estimate; the intermediate factors of $r$ are handled by the uniform splitting estimate. Restricting the factorization places all possible defects in controlled ranges.

🏷️ Prime extraction in a progression

The candidate weights mark prime divisors of the predecessor. A Type II estimate supplies bilinear cancellation for composite candidates, allowing a rough cofactor to be replaced by its local density. A least-prime-factor decomposition leaves a positive difference between the initial sifted mass and the removed composite mass.

The last balanced-semiprime range needs a separate upper bound. Its constants must be chosen independently of later marking parameters, so the previously obtained positive margin survives.

A progression condition on $mn$ is not necessarily invariant under removing marked factors. The source expands it in Dirichlet characters, separating it into factors on $m$ and $n$. It then verifies that these fixed twists preserve the required character and frequency estimates.

The marked estimate comes from the companion predecessor machinery. Consequently its coefficient restrictions and parameter order are part of the proof contract, not optional details.

🏷️ The conditional simultaneous companion

The second manuscript concerns distinct positive prime bases $q_1,\dots ,q_k$. It explicitly assumes four inputs:

• a uniform Hecke zero-free assertion;
• a marked Type II estimate;
• a block sieve;
• a rough-number density estimate.

Under those inputs, it proves a lower bound $c_{\mathbf{q}}x/(\log x{)}^2$ in every sufficiently large dyadic interval for primes at which all the $q_i$ are primitive roots. Their proofs in the companion literature are outside this second paper’s scope. (OpenAI, 2026b)

The progression makes every prescribed base a quadratic nonresidue. Splitting estimates then exclude the remaining index divisors for each base. Since the collection is fixed and finite, these exceptional sets can be combined.

This conditional interface should remain visible even though the first manuscript claims the relevant underlying results. The simultaneous paper does not itself provide independent proofs of its four inputs, and its theorem is stated for distinct positive prime bases, not every tuple of arbitrary admissible integers.

🏷️ Key steps and proof outlines

The source separates the proof into a supply of primes with controlled predecessors and a bound for the primes where the order is deficient. This separation lets the final argument be made quite explicit. The analytic zero-free theorem and marked bilinear estimates remain substantial inputs; the following explains their precise uses, not independent proofs of those results (OpenAI, 2026a).

A progression eliminates the quadratic obstruction

Write $a=d_0{t}^2$ with $d_0$ signed squarefree, and take $M=8{\prod }_{\ell \mid a,\ell \text{ odd}}\ell$. The progression lemma chooses a unit $u$ modulo $M$ such that $(a/p)=-1$ for primes $p\equiv u(\mathrm{mod}M)$, while also arranging

$$c\mid u-1,\gcd \left(\frac{u-1}{c},\frac{M}{c}\right)=1,c\in \{2,4\}.$$

Proof outline. At an odd prime divisor $j\ge 5$ of $d_0$, both squares and nonsquares remain available after excluding the residues zero and one. Quadratic reciprocity therefore allows one local choice to force the desired sign, while other local choices avoid divisibility of $u(u-1)$. If the squarefree part has no such divisor, its possibilities are $-1,\pm 2,\pm 3,\pm 6$ and the supplementary laws give explicit classes modulo $24$. The Chinese remainder theorem combines the choices. Selecting $u\equiv 3,7(\mathrm{mod}8)$ gives $c=2$, and $u\equiv 5(\mathrm{mod}8)$ gives $c=4$. Euler’s criterion then forces the order index to be odd.

Prime extraction preserves a positive remainder

The marked construction puts a nonnegative bounded weight $w(d)$ on candidates $d=crQ+1$ in the progression, with $Q$ prime. Write $X_0$ for its natural unsifted mass, $L=\log x$, and ${\mathfrak{S}}_M>0$ for its local-density product. A block sieve first keeps candidates with no prime factor at most $x^b$, giving normalized mass approximately $e^{-{\gamma }_E}/b$, where ${\gamma }_E$ is Euler’s constant.

To remove composites, assign each candidate its least prime factor $n$, place $\log n/L$ in a short bin, and write $d=mn$. The marked Type II estimate replaces the roughness condition on $m$ by a density $D_t(s)$, where $t$ is the prime cutoff exponent and $s=\log m/L$. Its normalization is selected so that

$${\int }_b^{1/2}{D}_t(1-t)\frac{dt}{t}=D_b(1)-1.$$

The $-1$ is the decisive surviving mass: the initial sieve main term is close to $D_b(1)$, and the total least-factor removal costs close to $D_b(1)-1$. A fixed fine partition turns this integral comparison into the required finite sum of estimates, with error less than $0.1$ in the source’s normalization.

Composites whose factors are both near $\sqrt{x}$ need a separate argument. In each dyadic box, replace the two prime indicators successively by rough-number densities. A two-variable upper sieve bounds the resulting pairs; its local forbidden conditions are $mn\equiv 1(\mathrm{mod}\ell )$ and, at smaller primes, $mn\equiv 0(\mathrm{mod}\ell )$. These occupy respectively $\ell -1$ and $2\ell -1$ disjoint residue pairs. Summing the boxes costs at most $C_2(\kappa +L^{-1})X_0/L$ when their factors exceed $x^{1/2-\kappa }$.

The order of choices now matters: fix the progression and hence $C_2$, choose $\kappa$ so this cost is small, choose the sieve exponent $b$, fix the bins, and only afterward choose the number of marking groups required by the finitely many Type II estimates. The final normalized lower bound is

$$0.9-0.2-\frac{1}{4}+o(1)>\frac{1}{4}.$$

Since $X_0\gg x/L$ and the fixed marking construction gives a uniform pointwise bound on $w$, positive weighted mass gives $\gg x/L^2$ distinct prime candidates.

Smoothing makes the splitting estimate uniform

For a possible odd index divisor $q$, the source uses the splitting-field interface described above. The relevant field has degree $n=q(q-1)$ and discriminant $D$ satisfying $\log D+n{\ll }_{a}n\log (2q)$. Its zero-free region is obtained from the separate Hecke theorem through abelian extensions; these field-theoretic facts are used here as stated inputs.

The analytic calculation takes a nonnegative smooth $f$ supported away from zero with $f\ge 1$ on $[1,2]$. Its Mellin transform $\Phi$ decays faster than any power on fixed vertical strips. Mellin inversion of the logarithmic derivative of the field’s zeta function, followed by shifting the contour, gives a prime-ideal sum with pole contribution $x\Phi (1)$ and zero contributions $-x^{\rho }\Phi (\rho )$. A uniform unit-height zero count is $O(\log D+n\log (2+|T|))$. Summing it against the rapid decay of $\Phi$ and using $\Re \rho \le 1-{\delta }_0$ yields

$${\Theta }_{K,f}(x){\ll }_{f}x+x^{1-{\delta }_0}(\log D+n).$$

A completely split rational prime in $(x,2x]$ contributes at least $nL$ to this nonnegative sum. Dividing by $nL$ therefore gives

$$\#\{\text{split primes in }(x,2x]\}{\ll }_a\frac{x}{q(q-1)L}+x^{1-{\delta }_0},$$

uniformly for $q\le \exp (L^{0.3})$. Keeping degree and discriminant dependence explicit is what permits summation over a growing range of $q$.

Both exceptional sets are negligible

For the large factor $Q>x^{0.9}$, an order defect implies $p\mid a^j-1$ for some $j=(p-1)/Q<2x^{0.1}$. Because $|a|>1$, the product of all these nonzero integers has logarithm $O_a(x^{0.2})$. Each distinct divisor $p>x$ consumes at least $L$ of that logarithm, so there are only $O_a(x^{0.2}/L)$ exceptional primes.

For a prime divisor $q$ of $r$, sum the splitting estimate over $\exp (L^{0.1})<q<\exp (L^{0.3})$. The main terms are bounded by the telescoping integer sum of $1/(q(q-1))$, giving total loss

$$O_a\left(\frac{x}{L}{e}^{-L^{0.1}}+x{e}^{-{\delta }_{0}L+L^{0.3}}\right)=o(x/L^2).$$

Thus both losses are smaller than the constructed supply. In the simultaneous companion, quadratic reciprocity and one Chinese-remainder progression make each of the finitely many prime bases a nonresidue; a finite union of these exceptional sets remains $o(x/L^2)$. That argument is conditional on its four stated inputs throughout. It does not assert a result uniform in a growing number of bases (OpenAI, 2026b).

🏷️ Connections

Zero-free half-planes explains reciprocal Dirichlet series and analytic continuation. Prime predecessor factorization supplies the graph and marked-bilinear background needed for the controlled-prime construction.

🏷️ Manuscripts and verification scope

• Primitive roots for every admissible integer base — pinned manuscript and companion source.
• Simultaneous primitive roots: a conditional lower bound for prime bases — pinned manuscript and companion source.

No family scope page was present at lean/docs/029.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. Primitive roots for every admissible integer base.

🐻  OpenAI 2026b. Simultaneous primitive roots: a conditional lower bound for prime bases.