Pairs of square-free values of the type $n^2+1$, $n^2+2$

Stoyan Dimitrov

Displaying similar documents to “Pairs of square-free values of the type $n^{2} + 1$ , $n^{2} + 2$ ”

Consecutive square-free values of the type $x^{2} + y^{2} + z^{2} + k$ , $x^{2} + y^{2} + z^{2} + k + 1$

Ya-Fang Feng (2023)

Czechoslovak Mathematical Journal

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We show that for any given integer $k$ there exist infinitely many consecutive square-free numbers of the type $x^{2} + y^{2} + z^{2} + k$ , $x^{2} + y^{2} + z^{2} + k + 1$ . We also establish an asymptotic formula for $1 \leq x, y, z \leq H$ such that $x^{2} + y^{2} + z^{2} + k$ , $x^{2} + y^{2} + z^{2} + k + 1$ are square-free. The method we used in this paper is due to Tolev.

On fixed point free involutions of $R^{1} \times S^{2}$

Gerhard X. Ritter (1976)

Colloquium Mathematicae

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On the $r$ -free values of the polynomial $x^{2} + y^{2} + z^{2} + k$

Gongrui Chen, Wenxiao Wang (2023)

Czechoslovak Mathematical Journal

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Let $k$ be a fixed integer. We study the asymptotic formula of $R (H, r, k)$ , which is the number of positive integer solutions $1 \leq x, y, z \leq H$ such that the polynomial $x^{2} + y^{2} + z^{2} + k$ is $r$ -free. We obtained the asymptotic formula of $R (H, r, k)$ for all $r \geq 2$ . Our result is new even in the case $r = 2$ . We proved that $R (H, 2, k) = c_{k} H^{3} + O (H^{9 / 4 + ε})$ , where $c_{k} > 0$ is a constant depending on $k$ . This improves upon the error term $O (H^{7 / 3 + ε})$ obtained by G.-L. Zhou, Y. Ding (2022).

Equalizers and coactions of groups

Martin Arkowitz, Mauricio Gutierrez (2002)

Fundamenta Mathematicae

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If f:G → H is a group homomorphism and p,q are the projections from the free product G*H onto its factors G and H respectively, let the group $_{f} \subseteq G * H$ be the equalizer of fp and q:G*H → H. Then p restricts to an epimorphism $p_{f} {= p |}_{f} :_{f} \to G$ . A right inverse (section) $G \to_{f}$ of $p_{f}$ is called a coaction on G. In this paper we study $_{f}$ and the sections of $p_{f}$ . We consider the following topics: the structure of $_{f}$ as a free product, the restrictions on G resulting from the existence of a coaction, maps of coactions and...

On $k$ -free numbers over Beatty sequences

Wei Zhang (2023)

Czechoslovak Mathematical Journal

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We consider $k$ -free numbers over Beatty sequences. New results are given. In particular, for a fixed irrational number $α > 1$ of finite type $τ < \infty$ and any constant $ε > 0$ , we can show that $\sum_{\binom{1 \leq n \leq x}{[α n + β] \in 𝒬_{k}}} 1 - \frac{x}{ζ (k)} ≪ x^{k / (2 k - 1) + ε} + x^{1 - 1 / (τ + 1) + ε},$ where $𝒬_{k}$ is the set of positive $k$ -free integers and the implied constant depends only on $α,$ $ε,$ $k$ and $β .$ This improves previous results. The main new ingredient of our idea is employing double exponential sums of the type $\sum_{1 \leq h \leq H} \sum_{\binom{1 \leq n \leq x}{n \in 𝒬_{k}}} e (ϑ h n) .$

The range of non-linear natural polynomials cannot be context-free

Dömötör Pálvölgyi (2020)

Kybernetika

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Suppose that some polynomial $f$ with rational coefficients takes only natural values at natural numbers, i. e., $L = {f (n) ∣ n \in ℕ} \subseteq ℕ$ . We show that the base- $q$ representation of $L$ is a context-free language if and only if $f$ is linear, answering a question of Shallit. The proof is based on a new criterion for context-freeness, which is a combination of the Interchange lemma and a generalization of the Pumping lemma.

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...

Prescribing endomorphism algebras of $ℵ_{n}$ -free modules

Rüdiger Göbel, Daniel Herden, Saharon Shelah (2014)

Journal of the European Mathematical Society

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It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass [1] and Eklof [8], both jointly with Shelah. In the first case...

Spreading and vanishing in nonlinear diffusion problems with free boundaries

Yihong Du, Bendong Lou (2015)

Journal of the European Mathematical Society

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We study nonlinear diffusion problems of the form $u_{t} = u_{x x} + f (u)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special $f (u)$ of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any $f (u)$ which is $C^{1}$ and satisfies $f (0) = 0$ , we show that the omega limit set $ω (u)$ of every bounded positive solution is determined by a stationary...

On the distribution of $(k, r)$ -integers in Piatetski-Shapiro sequences

Teerapat Srichan (2021)

Czechoslovak Mathematical Journal

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A natural number $n$ is said to be a $(k, r)$ -integer if $n = a^{k} b$ , where $k > r > 1$ and $b$ is not divisible by the $r$ th power of any prime. We study the distribution of such $(k, r)$ -integers in the Piatetski-Shapiro sequence ${⌊ n^{c} ⌋}$ with $c > 1$ . As a corollary, we also obtain similar results for semi- $r$ -free integers.

On operators from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ or to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$

Christian Samuel (2010)

Colloquium Mathematicae

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We show that every operator from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ is compact when 1 ≤ p,q < s and that every operator from $ℓ_{s}$ to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$ is compact when 1/p + 1/q > 1 + 1/s.

Coleff-Herrera currents, duality, and noetherian operators

Mats Andersson (2011)

Bulletin de la Société Mathématique de France

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Let $ℐ$ be a coherent subsheaf of a locally free sheaf $𝒪 (E_{0})$ and suppose that $ℱ = 𝒪 (E_{0}) / ℐ$ has pure codimension. Starting with a residue current $R$ obtained from a locally free resolution of $ℱ$ we construct a vector-valued Coleff-Herrera current $μ$ with support on the variety associated to $ℱ$ such that $φ$ is in $ℐ$ if and only if $μ φ = 0$ . Such a current $μ$ can also be derived algebraically from a fundamental theorem of Roos about the bidualizing functor, and the relation between these two approaches is discussed....

Product decompositions of quasirandom groups and a Jordan type theorem

Nikolay Nikolov, László Pyber (2011)

Journal of the European Mathematical Society

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We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$ , then for every subset $B$ of $G$ with $| B | > | G | / k^{1 / 3}$ we have $B^{3} = G$ . We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if $k \geq 2$ , then $G$ has a...

An application of the method wherein the $p h i$ - and $c h i$ - methods are combined for the determination of the grating constant. [II.]

Swami Jnanananda (1936)

Časopis pro pěstování matematiky a fysiky

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Maximum bipartite subgraphs in $H$ -free graphs

Jing Lin (2022)

Czechoslovak Mathematical Journal

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Given a graph $G$ , let $f (G)$ denote the maximum number of edges in a bipartite subgraph of $G$ . Given a fixed graph $H$ and a positive integer $m$ , let $f (m, H)$ denote the minimum possible cardinality of $f (G)$ , as $G$ ranges over all graphs on $m$ edges that contain no copy of $H$ . In this paper we prove that $f (m, θ_{k, s}) \geq \frac{1}{2} m + Ω (m^{(2 k + 1) / (2 k + 2)})$ , which extends the results of N. Alon, M. Krivelevich, B. Sudakov. Write $K_{k}^{'}$ and $K_{t, s}^{'}$ for the subdivisions of $K_{k}$ and $K_{t, s}$ . We show that $f (m, K_{k}^{'}) \geq \frac{1}{2} m + Ω (m^{(5 k - 8) / (6 k - 10)})$ and $f (m, K_{t, s}^{'}) \geq \frac{1}{2} m + Ω (m^{(5 t - 1) / (6 t - 2)})$ , improving a result of Q. Zeng, J. Hou. We also give lower bounds on...

The existence of $P (α)$ -points of N* for $ℵ_{0} α < 𝔠$

A. Szymański (1977)

Colloquium Mathematicae

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Displaying similar documents to “Pairs of square-free values of the type n 2 + 1 , n 2 + 2 ”

Displaying similar documents to “Pairs of square-free values of the type $n^{2} + 1$ , $n^{2} + 2$ ”