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PRIMES is in P
 Ann. of Math
, 2002
"... We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1 ..."
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We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1
Bounded gaps between primes with a given primitive root, II
"... Let m be a natural number, and letQ be a set containing at least exp(Cm) primes. We show that one can find infinitely many strings of m consecutive primes each of which has some q ∈ Q as a primitive root, all lying in an interval of length OQ(exp(C ′m)). This is a bounded gaps variant of a theorem o ..."
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Let m be a natural number, and letQ be a set containing at least exp(Cm) primes. We show that one can find infinitely many strings of m consecutive primes each of which has some q ∈ Q as a primitive root, all lying in an interval of length OQ(exp(C ′m)). This is a bounded gaps variant of a theorem of Gupta and Ram Murty. We also prove a result on an elliptic analogue of Artin’s conjecture. Let E/Q be an elliptic curve with an irrational 2torsion point. Assume GRH. Then for every m, there are infinitely many strings of m consecutive primes p for which E(Fp) is cyclic, all lying an interval of length OE(exp(C ′′m)). If E has CM, then the GRH assumption can be removed. Here C, C ′, and C ′ ′ are absolute constants.
THE EUCLIDEAN ALGORITHM FOR NUMBER FIELDS AND PRIMITIVE ROOTS
"... In 1927 Artin formulated his famous conjecture about primitive roots. Artin’s Primitive Root Conjecture. If a is not1 or a square then there are infinitely many primes p such that a is a primitive root modulo p. In fact, for an explicit constant A(a) Artin conjectured that the number of primes p ≤ ..."
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In 1927 Artin formulated his famous conjecture about primitive roots. Artin’s Primitive Root Conjecture. If a is not1 or a square then there are infinitely many primes p such that a is a primitive root modulo p. In fact, for an explicit constant A(a) Artin conjectured that the number of primes p ≤ x such that a is a