Artin's conjecture on primitive roots

Artin's conjecture predicts that every integer other than −1 or a square is a primitive root modulo infinitely many primes.

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Primitive root modulo n: An integer whose powers generate every invertible residue class modulo n. The conjecture asks how often a fixed integer has this property modulo a prime.
Fermat's little theorem: For prime p not dividing a, a^(p−1) is congruent to 1 modulo p. It bounds every nonzero residue's order by p−1.
Emil Artin: A mathematician whose work spanned algebra, number theory, and the theory of fields. He formulated the primitive-root conjecture in 1927.
Diffie–Hellman key exchange: A protocol that lets two parties establish a shared secret using exponentiation in a finite group. Primitive roots can generate the multiplicative group used in classic finite-field versions.
Generalized Riemann hypothesis for Dedekind zeta functions: The conjecture that nontrivial zeros of Dedekind zeta functions lie on the critical line. Hooley's conditional result depends on a related hypothesis for number fields arising in the proof.
Multiplicative order: The least positive exponent that makes a given invertible residue class equal to 1 modulo n. A number is a primitive root precisely when its multiplicative order is maximal.
Lagrange's theorem: In a finite group, the order of each subgroup divides the order of the group. It explains why multiplicative orders divide p−1.
Helmut Hasse: A German mathematician known for major contributions to algebraic number theory. His correspondence with Artin helped establish the conjecture's early context.
Finite field: A field containing finitely many elements. Nonzero elements of a prime field form the group in which primitive roots operate.
Artin's conjecture for average values: Results showing Artin-type primitive-root behavior on average over choices of the integer. Average results provide evidence without settling the conjecture for each fixed admissible integer.