Resolving Grosswald's conjecture on GRH

McGown, Kevin J.; Treviño, Enrique; Trudgian, Tim

doi:10.7169/facm/2016.55.2.5

Cited by 12 publications

(12 citation statements)

References 10 publications

(8 reference statements)

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“…For example, we are able to show that g (p) < p 3/4 for all p > 1.2 · 10 34 . We also note that in (5) we have used the Pólya-Vinogradov inequality M <n≤N…”

Section: Application Of Theoremmentioning

confidence: 99%

“…We note that using (1) does not allow one to show that g (p) ≪ p 1/2 . However, based on computational evidence, the bound in (2) and recent work in [2,5] it seems reasonable to extrapolate, as below.…”

Section: Introductionmentioning

confidence: 99%

“…Letĝ(p) denote the least prime primitive root modulo p. It is not known whetherĝ(p) < p for all p, or even for all sufficiently large p. The best unconditional result is due to Ha [3], namely, thatĝ(p) ≪ p 3.1 . On the Generalised Riemann Hypothesis it is known [7] that g(p) ≪ (log p) 6+ǫ , and, recently, it was shown in [5] thatĝ(p) < √ p − 2 for all p > 2791.…”

Section: Introductionmentioning

confidence: 99%

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On the least square-free primitive root modulo p

Cohen

Trudgian

2017

Journal of Number Theory

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“…For example, we are able to show that g (p) < p 3/4 for all p > 1.2 · 10 34 . We also note that in (5) we have used the Pólya-Vinogradov inequality M <n≤N…”

Section: Application Of Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the least square-free primitive root modulo p

Cohen

Trudgian

2017

Journal of Number Theory

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“…Rearranging (17) gives a lower bound on q, which, when combined with (16) gives us a hybrid lower bound for q. Taking the minimum over all possible values of m gives us the following…”

Section: The Modified Prime Sievementioning

confidence: 99%

Primitive elements with prescribed traces

Booker¹,

Cohen²,

Leong³

et al. 2021

Preprint

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Given a prime power q and a positive integer n, let F q n denote the finite field with q n elements. Also let a, b be arbitrary members of the ground field F q . We investigate the existence of a non-zero element ξ ∈ F q n such that ξ + ξ −1 is primitive and T (ξ) = a, T (ξ −1 ) = b, where T (ξ) denotes the trace of ξ in F q . This was a question intended to be addressed by Cao and Wang in 2014. Their work dealt instead with another problem already in the literature. Our solution deals with all values of n ≥ 5.A related study involves the cubic extension F q 3 of F q . We show that when either q ≥ 8 • 10 12 or q 3 − 1 has at least 24 distinct prime factors, then, for any a ∈ F q we can find a primitive element ξ ∈ F q 3 such that ξ + ξ −1 is also a primitive element of F q 3 , and for which the trace of ξ is equal to a. The improves a result of Cohen and Gupta. Along the way we prove a hybridised lower bound on prime divisors in various residue classes, which may be of interest to related existence questions.

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“…However, since we only know that p − 1 ≥ p 1 · · · p 9 > 2.2 · 10 8 we still have some cases to check. We proceed according to the 'divide and conquer' scheme of [5].…”

Section: Proof Of the Existence Theoremmentioning

confidence: 99%

Lehmer numbers and primitive roots modulo a prime

Cohen

Trudgian

2019

Journal of Number Theory

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A Lehmer number modulo a prime p is an integer a with 1 ≤ a ≤ p − 1 whose inversē a within the same range has opposite parity. Lehmer numbers that are also primitive roots have been discussed by Wang and Wang [7] in an endeavour to count the number of ways 1 can be expressed as the sum of two primitive roots that are also Lehmer numbers (an extension of a question of S. Golomb). In this paper we give an explicit estimate for the number of Lehmer primitive roots modulo p and prove that, for all primes p = 2, 3, 7, Lehmer primitive roots exist. We also make explicit the known expression for the number of Lehmer numbers modulo p and improve the Wang-Wang estimate for the number of solutions to the Golomb-Lehmer primitive root problem.

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Resolving Grosswald's conjecture on GRH

Cited by 12 publications

References 10 publications

On the least square-free primitive root modulo p

On the least square-free primitive root modulo p

Primitive elements with prescribed traces

Lehmer numbers and primitive roots modulo a prime

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