On Grosswald's conjecture on primitive roots

Stephen D. Cohen; Tomás Oliveira e Silva; Tim Trudgian

Displaying similar documents to “On Grosswald's conjecture on primitive roots”

Average r-rank Artin conjecture

Lorenzo Menici, Cihan Pehlivan (2016)

Acta Arithmetica

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Let Γ ⊂ ℚ * be a finitely generated subgroup and let p be a prime such that the reduction group Γₚ is a well defined subgroup of the multiplicative group ₚ*. We prove an asymptotic formula for the average of the number of primes p ≤ x for which [ₚ*:Γₚ] = m. The average is taken over all finitely generated subgroups $Γ = ⟨ a ₁, . . ., a_{r} ⟩ \subset ℚ *$ , with $a_{i} \in ℤ$ and $a_{i} \leq T_{i}$ , with a range of uniformity $T_{i} > e x p (4 {(l o g x l o g l o g x)}^{1 / 2})$ for every i = 1,...,r. We also prove an asymptotic formula for the mean square of the error terms in the asymptotic formula with...

Extremely primitive groups and linear spaces

Haiyan Guan, Shenglin Zhou (2016)

Czechoslovak Mathematical Journal

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A non-regular primitive permutation group is called extremely primitive if a point stabilizer acts primitively on each of its nontrivial orbits. Let $𝒮$ be a nontrivial finite regular linear space and $G \leq Aut (𝒮) .$ Suppose that $G$ is extremely primitive on points and let rank $(G)$ be the rank of $G$ on points. We prove that rank $(G) \geq 4$ with few exceptions. Moreover, we show that $Soc (G)$ is neither a sporadic group nor an alternating group, and $G = PSL (2, q)$ with $q + 1$ a Fermat prime if $Soc (G)$ is a finite classical simple group. ...

On the generalized vanishing conjecture

Zhenzhen Feng, Xiaosong Sun (2019)

Czechoslovak Mathematical Journal

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We show that the GVC (generalized vanishing conjecture) holds for the differential operator $Λ = (\partial_{x} - Φ (\partial_{y})) \partial_{y}$ and all polynomials $P (x, y)$ , where $Φ (t)$ is any polynomial over the base field. The GVC arose from the study of the Jacobian conjecture.

Results related to Huppert’s $ρ$ - $σ$ conjecture

Xia Xu, Yong Yang (2023)

Czechoslovak Mathematical Journal

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We improve a few results related to Huppert’s $ρ$ - $σ$ conjecture. We also generalize a result about the covering number of character degrees to arbitrary finite groups.

On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function

Ke-Pao Lin, Xue Luo, Stephen S.-T. Yau, Huaiqing Zuo (2014)

Journal of the European Mathematical Society

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It is well known that getting the estimate of integral points in right-angled simplices is equivalent to getting the estimate of Dickman-De Bruijn function $ψ (x, y)$ which is the number of positive integers $\leq x$ and free of prime factors $> y$ . Motivating from the Yau Geometry Conjecture, the third author formulated the Number Theoretic Conjecture which gives a sharp polynomial upper estimate that counts the number of positive integral points in n-dimensional ( $n \geq 3$ ) real right-angled simplices. In this...

On the Brocard-Ramanujan problem and generalizations

Andrzej Dąbrowski (2012)

Colloquium Mathematicae

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Let $p_{i}$ denote the ith prime. We conjecture that there are precisely 28 solutions to the equation $n ² - 1 = p ₁^{α ₁} \dots p_{k}^{α_{k}}$ in positive integers n and α₁,..., $α_{k}$ . This conjecture implies an explicit description of the set of solutions to the Brocard-Ramanujan equation. We also propose another variant of the Brocard-Ramanujan problem: describe the set of solutions in non-negative integers of the equation n! + A = x₁²+x₂²+x₃² (A fixed).

Growth of a primitive of a differential form

Jean-Claude Sikorav (2001)

Bulletin de la Société Mathématique de France

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For an exact differential form on a Riemannian manifold to have a primitive bounded by a given function $f$ , by Stokes it has to satisfy some weighted isoperimetric inequality. We show the converse up to some constants if $M$ has bounded geometry. For a volume form, it suffices to have the inequality ( $| Ω | \leq \int_{\partial Ω} f d σ$ for every compact domain $Ω \subset M$ ). This implies in particular the “well-known” result that if $M$ is the universal covering of a compact Riemannian manifold with non-amenable fundamental group, then...

On the distribution of consecutive square-free primitive roots modulo $p$

Huaning Liu, Hui Dong (2015)

Czechoslovak Mathematical Journal

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A positive integer $n$ is called a square-free number if it is not divisible by a perfect square except $1$ . Let $p$ be an odd prime. For $n$ with $(n, p) = 1$ , the smallest positive integer $f$ such that $n^{f} \equiv 1 (mod p)$ is called the exponent of $n$ modulo $p$ . If the exponent of $n$ modulo $p$ is $p - 1$ , then $n$ is called a primitive root mod $p$ . Let $A (n)$ be the characteristic function of the square-free primitive roots modulo $p$ . In this paper we study the distribution $\sum_{n \leq x} A (n) A (n + 1),$ and give an asymptotic formula by using properties of character...

A geometric construction for spectrally arbitrary sign pattern matrices and the $2 n$ -conjecture

Dipak Jadhav, Rajendra Deore (2023)

Czechoslovak Mathematical Journal

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We develop a geometric method for studying the spectral arbitrariness of a given sign pattern matrix. The method also provides a computational way of computing matrix realizations for a given characteristic polynomial. We also provide a partial answer to $2 n$ -conjecture. We determine that the $2 n$ -conjecture holds for the class of spectrally arbitrary patterns that have a column or row with at least $n - 1$ nonzero entries.

On coarse embeddability into $ℓ_{p}$ -spaces and a conjecture of Dranishnikov

Piotr W. Nowak (2006)

Fundamenta Mathematicae

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We show that the Hilbert space is coarsely embeddable into any $ℓ_{p}$ for 1 ≤ p ≤ ∞. It follows that coarse embeddability into ℓ₂ and into $ℓ_{p}$ are equivalent for 1 ≤ p < 2.