Artin’s primitive root conjecture

Repeatedly multiply by 2 and keep only the remainder after division by 5. The remainders are 2, 4, 3, 1, then the cycle repeats: every nonzero remainder appears. Division by 7 gives 2, 4, 1, then repeats, missing 3, 5 and 6. Artin's conjecture asks whether one fixed eligible integer succeeds like the first example for a predicted positive share of primes. The integer can be any integer other than −1 or a perfect square, and primes dividing it are excluded.[1]

Fix an integer and multiply its successive powers modulo changing primes: how often does that one integer generate every nonzero remainder? Artin's conjecture predicts both a positive limiting proportion and its value. The fixed integer must be neither −1 nor a perfect square; primes dividing it are excluded.

For $a=2$ and $p=5$, the powers give remainders 2, 4, 3 and 1. For $p=7$, they give 2, 4 and 1. The multiplicative order is the first positive exponent returning the remainder 1; here it is 4 and 3 respectively. A primitive root has order $p-1$, the number of nonzero remainders.

To define density, count successful primes up to a cutoff x and divide by $\pi(x)$, the number of all primes up to x. The conjecture predicts the limit of this fraction as x grows. The constant is not automatically the same for every eligible integer: algebraic dependencies can change the predicted proportion.[1]

The predicted frequency has not been proved without extra assumptions.[1]

The density formula is known under a generalized Riemann hypothesis. The August 2026 paper instead assumes a non-abelian sieve inequality to prove infinitely many successful primes; its stated theorem is for 2, with extension to eligible integers discussed separately. Infinitely many successes is weaker than the full limiting-density formula. Neither establishes this page's unconditional target.[1] [2]

For the fixed integer 10, the same question governs how often reciprocals of primes have the longest possible repeating decimal. For example, one seventh repeats 142857, a six-digit block. It connects a familiar decimal pattern with the distribution of primes.[2]

Long division makes the connection to decimal expansions precise. For a prime p other than 2 or 5, the period of $1/p$ is the multiplicative order of 10 modulo p. If 10 is a primitive root, the period is the maximum $p-1$. Thus the conjecture for 10 would predict how often this maximal decimal pattern occurs. More generally, it connects finite multiplication cycles to the arithmetic of number fields used to predict which primes exhibit them.[1] [2]