Primitive roots for every admissible integer base

Every whole number except −1 and perfect squares is claimed to be a primitive root for infinitely many primes.

2 papers October 4, 2026 Explained by AI

For a prime p, a number a is a primitive root when its powers run through every nonzero remainder mod p. In 1927 Artin conjectured that every suitable base does this for infinitely many primes. The first paper claims to prove that infinitude part for every base that is not −1 and not a square. It gives a lower count of such primes, below the size Artin predicted. A second paper claims the same for several prime bases at once, assuming four inputs from the first.

First step

1927

E. Artin, who formulated the primitive root conjecture

Significance

Landmark

The infinitude part of Artin's conjecture was open for every single base, including 2, if this proof holds.

Trust

Claimed, not checked

Field

Number theory

2 papers

The explanation

Take the prime 5 and keep doubling, keeping only the remainder after dividing by 5. You get 2, 4, 3, 1. Every nonzero remainder appears before the cycle repeats. So 2 is a primitive root mod 5 (its powers reach every nonzero remainder). Try the prime 7 instead. Doubling gives 2, 4, 1, and the cycle restarts after three steps. So 2 is not a primitive root mod 7. Among primes below 60, base 2 works for 3, 5, 11, 13, 19, 29, 37, 53 and 59.

Some bases can never do this for large primes. Powers of −1 only reach 1 and −1. A perfect square such as 4 always falls short too. Mod 5, its powers are 4 and 1. The paper calls the other integers admissible: not −1 and not a squareSource 1. In 1927 Artin conjectured that every admissible integer is a primitive root for infinitely many primesSource 2. He also predicted what share of primes it should be.

Progress came in two forms. In 1967 Hooley proved Artin’s predicted count, but only if a version of the Riemann hypothesis holdsSource 3. That hypothesis is still unproved. Without it, results only handled a short list of bases at once. Heath-Brown proved in 1986 that at least one of 2, 3 and 5 worksSource 4. Results on average over many bases also exist. But the exceptions there can shift as the range grows, so they do not settle any one baseSource 5.

The first paper claims the infinitude part for every admissible base, including 2Source 6. For each base, it claims at least a fixed multiple of x/(log x)² such primes between x and 2xSource 7. Here log is the natural logarithm. At x = 1,000,000, x/(log x)² is about 5,239. The predicted size, x/log x, is about 72,382. So the count is smaller than Artin’s prediction, but it never runs out. The constant depends on the base.

The paper names two obstaclesSource 8. Whether a works for p depends on hidden factors of p − 1. Each possible factor brings its own number field (a number system that extends the fractions). One obstacle is analytic. The paper needs one uniform zero-free region for a whole growing family of L-functions (functions built from primes in those fields). Take cyclotomic fields (number fields built from roots of unity) that contain the 12th roots of unity. For these, it claims no zeros where the real part of the input s is above 1 − 10⁻⁶Source 9. The same width serves every such field and characterSource 10. Standard zero-free regions instead get thinner as the field growsSource 11. The other obstacle is to build many primes p whose p − 1 has factors only in ranges this strip controls.

The argument leans on companion papers in the same release. The prime construction uses an estimate from a paper on the prime factors of p − 1Source 12. The analytic part builds on another companion paper on zero-free regionsSource 13. A second paper claims a version for several prime bases at onceSource 14. It assumes four inputs from the first paper and does not prove themSource 15.

How sure is this? No Lean formalization is listed for these papers in the openai/math release. So no computer proof checker has checked them. The result rests on new zero-free regions and on companion papers. No specialist review has been published.

What it doesn't show. This does not prove Artin's predicted share of primes, only a smaller lower count.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Landmark only if the proof holds, including its new zero-free region and the companion papers it uses. No Lean formalization is listed and no specialist review is published. The count is below Artin's predicted share.

Words used

Primitive root: A number whose powers, taken mod a prime p, reach every nonzero remainder before repeating.
Admissible base: An integer that is neither −1 nor a perfect square.
Number field: A number system that extends the fractions by adding roots of polynomial equations.
L-function: A function built from primes that encodes how they behave in a number field.
Zero-free region: A part of the complex plane where an L-function is shown to have no zeros.
Cyclotomic field: A number field made by adding roots of unity, the complex solutions of z^n = 1.
Riemann hypothesis: An unproved conjecture that places all nontrivial zeros of certain L-functions on one line.

The original papers (2)

Primitive roots for every admissible integer base (PDF, opens in a new tab)
October 4, 2026
Simultaneous primitive roots: a conditional lower bound for prime bases (PDF, opens in a new tab)
October 4, 2026

All quoted lines (15)

Call $a$ admissible if $a\ne-1$ and $a$ is not the square of an integer. These are necessary conditions for primitive-root behavior at infinitely many primes: the order of $-1$ is at most $2$ , and a nonzero square modulo an odd prime has order dividing $(p-1)/2$ .
Primitive roots for every admissible integer base Back to text
Artin's primitive root conjecture, formulated in 1927, asserts that every admissible integer is a primitive root modulo infinitely many primes; its quantitative form predicts a positive-density asymptotic.
Primitive roots for every admissible integer base Back to text
Hooley proved Artin's predicted asymptotic under the Riemann hypothesis for the Dedekind zeta functions of the Kummer fields $\Q(\mu_n,a^{1/n})$ , with $n$ squarefree.
Primitive roots for every admissible integer base Back to text
Heath-Brown proved that at most two prime bases fail the infinitude assertion, so at least one of $2,3,5$ satisfies it.
Primitive roots for every admissible integer base Back to text
The exceptional set may depend on $x$ , so these average results do not by themselves settle a prescribed base.
Primitive roots for every admissible integer base Back to text
In particular, it proves the infinitude assertion for each fixed admissible integer, including the prescribed base $2$ .
Primitive roots for every admissible integer base Back to text
We prove the infinitude assertion in Artin's primitive root conjecture: for every integer $a$ that is neither $-1$ nor a square, there are at least $c_a x/(\log x)^2$ primes in $(x,2x)$ with primitive root $a$ , for some $c_a>0$ and every sufficiently large $x$ .
Primitive roots for every admissible integer base Back to text
Two difficulties must be addressed for a fixed base.
Primitive roots for every admissible integer base Back to text
For every finite-order Hecke character $\eta$ of $F$ , the meromorphic continuation of $L_F(s,\eta)$ has no zeros in $\re s>1-10^{-6}.$
Primitive roots for every admissible integer base Back to text
The width is common to all fields and characters in the statement, with no conductor or height cutoff.
Primitive roots for every admissible integer base Back to text
A standard simultaneous zero-free region for Hecke characters associated with an abelian extension has logarithmic width depending on the field, conductors, and height, and permits at most one simple real exceptional zero.
Primitive roots for every admissible integer base Back to text
The marked prime estimate is an input from \emphThe Poisson–Dirichlet law for the prime factors of $p-1$ .
Primitive roots for every admissible integer base Back to text
The companion manuscript The Quasi-Riemann Hypothesis develops a coefficient reflection and an additive probe over that field while retaining unrestricted cube factors.
Primitive roots for every admissible integer base Back to text
For every fixed finite set of distinct positive primes, we prove that at least $cx/(\log x)^2$ primes in $(x,2x)$ have every member of the set as a primitive root, for some $c>0$ and all sufficiently large $x$ .
Simultaneous primitive roots: a conditional lower bound for prime bases Back to text
They are hypotheses of the following theorem; their proofs in and its cited companion manuscripts are outside the scope of this paper.
Simultaneous primitive roots: a conditional lower bound for prime bases Back to text

Other sources

028Uniformly bounded components of Gaussian-prime graphsNumber theory
026Positive lower density of large prime gapsNumber theory
024An asymptotic formula for the number of totientsNumber theory
023Patterson's first moment for cubic Gauss sumsNumber theory
022The weak inhomogeneous Duffin–Schaeffer conjectureNumber theory

Browse Number theory Prime numbers Open since before 1940 Claimed, not yet checked Landmark results