Primitive roots for every admissible integer base. 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 $c_a x/(\log x)^2$ primes in every sufficiently large interval $(x,2x)$ have primitive root a, with $c_a\gt 0$.
released 2026-10-04 | 3 theorems · 25 lemmas · 34 proofs · 38,435 words |
PLAY LEVEL 1 »(pdf)
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 $c_a\gt 0$ and every sufficiently large x.
released 2026-10-04 | 1 theorem · 3 lemmas · 5 proofs · 8,176 words |
PLAY LEVEL 2 »(pdf)
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. The result assumes four explicitly stated analytic and sieve inputs from the companion paper on primitive roots for admissible integer bases.