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  "body": "\n## Statement\n\nLet $a$ be an integer that is not a perfect square and is not $-1$.\nThen there are infinitely many [[prime numbers|01826KMG2H]] $p$ such that $a$ is a [[primitive root modulo|0F33KX9Q6F]] $p$.\n\n## Remarks\n\nThe conjecture is open for every single value of $a$.\nThe full conjecture also predicts a positive density for this set of primes; for many $a$ the predicted density is Artin's constant, about $0.3739558$.\nChristopher Hooley proved the conjecture in 1967 under the assumption of the [[generalized Riemann hypothesis|DNKA4WBKDS]].\nRoger Heath-Brown proved in 1986 that at most two primes $a$ fail the conjecture; so at least one of $2$, $3$, $5$ is a primitive root modulo infinitely many primes.\nThe two hypotheses are needed: if $a = b^2$ is a square and $p \\nmid a$, then $a^{(p-1)/2} \\equiv b^{p-1} \\equiv 1 \\pmod p$, so $a$ is never a primitive root modulo an odd prime $p$.\nFor $a = -1$, the [[multiplicative order|02RRHJ7R98]] of $-1$ is at most $2$, so it is a primitive root only modulo $2$ and $3$.\nFor example, $2$ is a primitive root modulo $3$, $5$, $11$, $13$, $19$ and $29$.\n\n## Explanations\n\n### L1 human\n\nA primitive root modulo $p$ is a number whose powers run through all nonzero remainders modulo $p$.\nArtin guessed that a fixed number such as $2$ is a primitive root for infinitely many primes, unless there is an obvious reason against it.\nSquares are such an obvious reason, and so is $-1$.\nThe guess is still unproved, even for $a = 2$.\n",
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      "first_stated": "1927",
      "posed_by": "Emil Artin"
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    "name": "Artin's conjecture on primitive roots",
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  "msc": [
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