Artin’s primitive roots for every admissible integer base

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Family 029 · Number theory
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

Take a prime $p$ and repeatedly multiply by an integer $a$, reducing modulo $p$ after each step. Sometimes this visits every nonzero residue before returning to one. For instance, the powers of two modulo five are $2,4,3,1$. In this case two is a primitive root modulo five.

Artin’s primitive-root conjecture asks whether a fixed suitable integer, such as two, has this property for infinitely many primes. The word “fixed” is essential. Knowing that every prime has some primitive root does not identify infinitely many primes for one prescribed base.

The first manuscript claims the infinitude statement for every admissible integer, with a lower bound in every sufficiently large interval $(x,2x)$. The second concerns several prescribed bases at once. It explicitly states its theorem as conditional on four analytic and sieve inputs from the first manuscript and its companions. Keeping these two logical formulations distinct is part of understanding the family.

2. Background and essential definitions

The nonzero residues modulo a prime form a cyclic multiplicative group $\mathbb F_p^\times$ of size $p-1$. If $p\nmid a$, the multiplicative order $\operatorname{ord}_p(a)$ is the least positive $k$ with $a^k\equiv1\pmod p$. Lagrange’s theorem says that this order divides $p-1$.

The base $a$ is a primitive root when its order is exactly $p-1$. Define its order index by

$$i_p(a)=\frac{p-1}{\operatorname{ord}_p(a)}.$$

The goal is $i_p(a)=1$. If the index is larger than one, some prime $\ell$ divides it, which is equivalent to the pair of conditions

$$p\equiv1\pmod\ell,\qquad a^{(p-1)/\ell}\equiv1\pmod p.$$

Thus every failure has a prime-divisor witness.

An integer is admissible here if it is neither $-1$ nor an integer square. These exclusions include zero and one. The order of $-1$ is at most two, and a square modulo an odd prime has order dividing $(p-1)/2$. Other negative integers are allowed by the first theorem.

Why number fields enter a modular-arithmetic problem

The field

$$\mathbb Q(\zeta_\ell,a^{1/\ell})$$

is obtained by adjoining an $\ell$th root of unity and an $\ell$th root of $a$. Away from the finitely many ramified primes, the two displayed congruences say that $p$ splits completely in this field: its prime ideal factors all have residue field $\mathbb F_p$. Counting order obstructions therefore becomes a problem about how primes split in varying number fields.

Hecke characters generalize residue-class characters to number fields. Their $L$-functions are Dirichlet series with Euler products encoding prime-ideal arithmetic. A zero-free half-plane gives analytic control over prime counts. Only the role of this machinery is needed below: uniform control as $\ell$ changes is crucial because all possible witnesses must be removed together.

3. Development history and previous partial results

Artin formulated the conjecture in 1927. Its quantitative version predicts a positive-density asymptotic among primes, with a constant depending on the base and its arithmetic obstructions.

Hooley proved that predicted asymptotic in 1967 under appropriate generalized Riemann hypotheses for the Dedekind zeta functions of the fields $\mathbb Q(\zeta_n,a^{1/n})$, with squarefree $n$. His strategy counts splitting conditions associated with divisors of the order index. The difficulty is controlling a growing collection of fields while keeping the accumulated errors small.

Unconditional progress allowed a finite choice of bases. Gupta and Murty produced a set of thirteen integers, formed from three distinct primes, at least one of which works infinitely often. Heath-Brown proved that at most two positive prime bases can fail the infinitude assertion. In particular, at least one of two, three and five works. This conclusion does not identify which member of that triple works, and does not settle the prescribed base two.

Average results take another direction. Goldfeld and Stephens established early theorems averaging over the base. More recent work of Klurman, Shparlinski and Teräväinen obtains the expected asymptotic for almost all bases in very short growing ranges. An exceptional set depending on the cutoff can still contain a particular fixed integer, so such an average theorem does not automatically settle each base.

For simultaneous primitive roots, Matthews obtained density results under generalized Riemann hypotheses, while Anwar and Pappalardi gave an infinitude criterion under Schinzel’s Hypothesis H. Those theorems concern several order conditions holding at the same primes.

The first new manuscript claims an unconditional fixed-base lower bound, weaker than the conjectured asymptotic by one logarithmic factor. The second transports its machinery to fixed finite sets of distinct positive prime bases, expressly retaining four stated inputs as hypotheses.

4. What the new manuscripts claim

M1: every admissible integer, fixed in advance

For every admissible integer $a$, M1 claims constants $c_a>0$ and $x_a$ such that for every real $x\ge x_a$,

$$\#\{p:x<p<2x,\ p\nmid a,\ \operatorname{ord}_p(a)=p-1\} \ge c_a\frac{x}{(\log x)^2}.$$

This gives infinitely many primes for each fixed base, including two. It does not give positive density among primes: dividing the lower bound by the prime-counting scale $x/\log x$ leaves a quantity tending to zero. The constants and starting point may depend on $a$.

M1 also claims an analytic theorem of independent scope. For every cyclotomic number field $F$ containing the twelfth roots of unity and every finite-order Hecke character $\eta$ of $F$,

$$L_F(s,\eta)\ne0 \qquad\text{when }\operatorname{Re}s>1-10^{-6}.$$

The principal character’s pole at one is permitted. There is no conductor or height cutoff, and the numerical width is the same for all these fields and characters. Auxiliary constants and thresholds in its proof may nevertheless depend on the fixed field and character. This theorem is not the Riemann hypothesis.

M2: simultaneous prime bases, under four inputs

Fix distinct positive primes $q_1,\ldots,q_k$, with $k$ finite. M2 claims, assuming its Inputs 3.1, 4.1, 5.1 and 5.2, at least

$$c_{\boldsymbol q}\frac{x}{(\log x)^2}$$

primes in every sufficiently large $(x,2x)$ for which every $q_i$ is a primitive root. The count excludes primes dividing the product of the bases.

The four inputs are the uniform Hecke zero-free assertion, a marked bilinear estimate for sieve coefficients, a block sieve estimate and a rough-number density estimate. “Marked” means that selected prime factors are required to occur in prescribed ranges. A rough integer has no small prime factors. These inputs are reproduced precisely in M2 and attributed to M1 and its cited companions; M2 does not claim to prove them internally.

This is an assertion about all prescribed prime bases working at the same primes, stronger than selecting one successful base from the set. It is not a statement for arbitrary finite collections of admissible integers, whose multiplicative relationships can create additional compatibility problems.

5. Why the advance matters

Importance: 5/5, provisionally. The fixed-base claim addresses a central classical problem about the multiplicative behavior of primes, including the long-standing case of base two. Its analytic theorem also has scope well beyond the immediate application.

The rating concerns the significance of the stated claims, not their verification. The shortfall from the predicted density remains substantial: infinitude with a quantitative lower bound is one part of Artin’s conjecture, not its full asymptotic form.

The simultaneous theorem adds another dimension. Several bases must all avoid smaller multiplicative subgroups at the same prime. By restricting to distinct positive prime bases, the manuscript can arrange their quadratic conditions together and then remove the remaining obstructions with a finite union of estimates.

6. The main idea

First restrict the factorization of the predecessor

Instead of examining all primes, M1 constructs many with

$$p=crQ+1,\qquad c\in\{2,4\},\qquad Q>x^{0.9}\text{ prime},$$

where every prime divisor of $r$ lies between

$$\exp((\log x)^{0.1}) \quad\text{and}\quad \exp((\log x)^{0.3}).$$

A fixed congruence condition makes $a$ a quadratic nonresidue modulo $p$. Consequently two cannot divide its order index. Every remaining prime divisor of the index must be either $Q$ or a prime factor of $r$. This separates the obstruction into two manageable ranges.

If $Q$ divides the index, then $p$ divides $a^j-1$ for some small $j=(p-1)/Q$. The product of these finitely many integers has too few large prime divisors to account for much of the constructed prime supply.

For a divisor of $r$, the complete-splitting interpretation applies. The common zero-free region provides a uniform estimate, and summing over the controlled range removes only a small fraction of the supply.

Obtain enough actual primes

Producing numbers of the displayed shape is easier than proving they are prime. The sieve starts with weighted candidates, removes small prime factors and then subtracts composites according to their least prime factor. Marked bilinear estimates control products of two variables with the required predecessor factors.

A separate estimate treats composites having two prime factors near the square root of the size. The parameters are chosen so that the total composite contribution leaves a positive remainder of primes. Fixed congruence conditions are incorporated using Dirichlet characters, which separate a condition on a product into factors on its two variables.

The manuscript explicitly uses marked estimates from the prime-predecessor work in family 011. Its zero-free construction develops mechanisms related to family 003 over general cyclotomic fields; the broader field scope is essential here, even though the stated numerical strip is narrower.

Compare two analytic views of the same average

For the zero-free theorem, M1 builds cubic theta coefficients with sextic twists. Theta series package arithmetic sums into objects with transformation laws. One transformation reduces relevant interactions to quadratic character sums, which a large-sieve inequality controls on average.

Poisson summation evaluates the same additive average differently. Its main frequency produces a reciprocal Hecke $L$-function, multiplied by a controlled nonvanishing factor. A Mellin transform, which converts scaling behavior into a complex variable, links the resulting power saving to continuation of this expression across the line $\operatorname{Re}s=1$. In the manuscript’s argument, that excludes zeros in the stated strip.

Impose several quadratic conditions together

M2 uses primes $p=4rQ+1$, with $rQ$ odd, so $p\equiv5\pmod8$. This makes two a quadratic nonresidue. For each prescribed odd prime base, quadratic reciprocity converts the desired nonresidue condition into a residue choice modulo that base. The Chinese remainder theorem combines the finitely many choices into one progression.

The same two-range removal then applies to each base. Since their number is fixed, the union of their exceptional sets remains smaller than the available prime supply.

7. Scope, limitations, and open questions

Neither manuscript establishes the predicted positive-density asymptotic or its exact constant. The lower bound has scale $x/(\log x)^2$, not $x/\log x$.

The base, or finite set of prime bases, is fixed before the counting parameter grows. No uniform result for an arbitrarily growing collection is stated. M1 allows all admissible integer bases, while M2’s simultaneous theorem is restricted to distinct positive primes and explicitly assumes its four inputs.

The analytic half-plane covers the specified cyclotomic fields and finite-order characters. It should not be replaced by a claim about every automorphic $L$-function or by the generalized Riemann hypothesis. Its uniform width also should not be confused with uniform auxiliary constants throughout the construction.

The digest treats the manuscripts’ mathematical statements as stated. In particular, the dependency structure and the conditional wording of M2 remain part of the result being explained.

8. Sources and further reading

Hooley’s On Artin’s conjecture is the primary source for the conditional asymptotic. Heath-Brown’s Artin’s conjecture for primitive roots explains the finite-choice breakthrough. For the modern average perspective, see Klurman, Shparlinski and Teräväinen, On Artin’s conjecture on average and short character sums.

M1 below states the every-base and Hecke zero-free claims. M2 states the simultaneous result with its four inputs explicitly identified. Their bibliographies include the earlier simultaneous-root work and the analytic and sieve sources used in the construction.

Source manuscripts

• M1. OpenAI, Primitive roots for every admissible integer base (October 4, 2026).
• M2. OpenAI, Simultaneous primitive roots: a conditional lower bound for prime bases (October 4, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.