A note on the least nonresidue modulo prime p≡7 (mod 8)
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In 2008, Andrew Granville proved that, Vinogradov's conjecture on the least nonresidue modulo primes p with p≡3 (mod 4) is true if every interval of length p near to p contains a p∈-smooth integer for all sufficiently large p. In this short note, we prove that it is enough to find a p∈-smooth integer in the interval [p-1/2 – p∈/2, p-1/2]. Further, when ∈ = 1/2, we provide a new proof of the Vinogradov's conjecture.
Abstract
In 2008, Andrew Granville proved that, Vinogradov's conjecture on the least nonresidue modulo primes p with p≡3 (mod 4) is true if every interval of length p near to p contains a p∈-smooth integer for all sufficiently large p. In this short note, we prove that it is enough to find a p∈-smooth integer in the interval [p-1/2 – p∈/2, p-1/2]. Further, when ∈ = 1/2, we provide a new proof of the Vinogradov's conjecture.
참고문헌 (8)
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