Primitive roots for every admissible integer base

1 · The essentials

Choose an integer that is neither −1 nor a square. For some prime numbers, repeatedly multiplying by that integer and taking remainders visits every nonzero remainder before repeating. When this happens, the integer is called a primitive root for that prime. The main manuscript claims that every allowed integer has this property for infinitely many primes. It also gives a minimum count of such primes between any sufficiently large number and twice that number. A companion manuscript claims a related result for several positive prime bases at once, subject to four additional assumptions. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine repeatedly taking the same kind of step around a circular arrangement of positions: the goal is to visit every allowed position before returning to the start. A primitive root achieves that full tour among the nonzero remainders, although its steps come from multiplication rather than ordinary movement.

Why it matters

The main claim addresses the infinitude assertion in Artin’s primitive root conjecture for every integer base covered by its stated restrictions. Its quantitative bound says more than simply that suitable primes never run out: it supplies a minimum count in every sufficiently large interval of the specified form. The companion claim asks for the stronger simultaneous property that several prescribed bases all work for the same primes, but it is conditional.

2 · Going further

For a prime $p$ , saying that $a$ is a primitive root means that its powers modulo $p$ run through all the nonzero residue classes. The main manuscript considers every integer $a$ that is neither $- 1$ nor a square. It claims that there is a constant $c_{a} > 0$ such that, for every sufficiently large $x$ , at least $c_{a} x / (lo g x)^{2}$ primes $p \in (x, 2 x)$ have $a$ as a primitive root. The constant is allowed to depend on the chosen base.

The manuscript identifies this as the infinitude assertion in Artin’s primitive root conjecture. The supplied abstract states the result but does not describe the main idea of its argument, so no proof strategy can be extracted from this source. In particular, the quantitative claim should be distinguished from a formula giving the exact frequency of suitable primes: no such formula is supplied here.

The companion manuscript treats every fixed finite set of distinct positive primes as bases. It claims that at least $c x / (lo g x)^{2}$ primes in $(x, 2 x)$ have every member of the set as a primitive root, with $c > 0$ and all sufficiently large $x$ . This simultaneous result assumes four explicitly stated analytic and sieve inputs from the main manuscript. Their contents are not included in the supplied abstracts, so the source establishes the stated dependency but does not allow those assumptions or their role in the argument to be examined.

Useful notions

Prime numbers
Remainders and modular arithmetic
Powers of an integer
Logarithms
Lower bounds and sufficiently large thresholds

What was known before / what is new

Before

The source places the main problem within Artin’s primitive root conjecture, specifically its infinitude assertion. It gives no account of earlier partial results or historical progress, so no comparison with prior bounds can be made from the supplied text.

New

The main manuscript claims a lower bound of $c_{a} x / (lo g x)^{2}$ for every integer base that is neither $- 1$ nor a square, in every sufficiently large interval $(x, 2 x)$ . This would give infinitely many suitable primes for each such base. The companion manuscript claims a lower bound of the same form for simultaneous primitive roots from a fixed finite set of distinct positive primes, conditional on four inputs.

Limits and assumptions

The main claim excludes −1 and squares, and its interval bound applies only for sufficiently large $x$ , with a positive constant depending on the base. The companion claim concerns fixed finite sets of distinct positive prime bases and assumes four analytic and sieve inputs whose statements are absent from the supplied text. The abstracts do not provide the argument, an exact frequency formula, or a description of what remains open beyond these claims. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Primitive root: An integer whose powers modulo a prime visit every nonzero residue class.
Residue class modulo a prime: A collection of integers that have the same remainder upon division by that prime.
Admissible integer base: In this family of manuscripts, an integer that is neither −1 nor a square.
Lower bound: A quantity that the count under consideration is asserted to be at least.
Simultaneous primitive roots: A set of bases each of which is a primitive root for the same prime.

Keywords : Primitive roots · Artin’s primitive root conjecture · Simultaneous primitive roots

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves the infinitude assertion in Artin's primitive root conjecture for every integer a that is neither −1 nor a square. For each such base, at least $c_{a} x / (lo g x)^{2}$ primes in every sufficiently large interval $(x, 2 x)$ have primitive root a, with $c_{a} > 0$ .

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

★

Primitive roots for every admissible integer base

04/10/2026

To be verified

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 $c_{a} x / (lo g x)^{2}$ primes in $(x, 2 x)$ with primitive root a, for some $c_{a} > 0$ and every sufficiently large x.

PDF GitHub folder

Simultaneous primitive roots: a conditional lower bound for prime bases

04/10/2026

To be verified

For every fixed finite set of distinct positive primes, we prove that at least $c x / (lo g x)^{2}$ primes in $(x, 2 x)$ have every member of the set as a primitive root, for some c > 0 and all sufficiently large x. The result assumes four explicitly stated analytic and sieve inputs from the companion paper on primitive roots for admissible integer bases.

PDF GitHub folder