Primes in the interval [2n,3n]

Bachraoui, Mohamed El

doi:10.12988/ijcms.2006.06065

Cited by 25 publications

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“…The proof of this Theorem is based on the the Bertrand's postulate which states that there exists a prime number in the interval [n, 2n] for all n ∈ Z + . This postulate was proved for the first time by P. L. Chebishev in 1850 and simplified later by P. Erdős in 1932 [3] due to M. El Bachraoui [1].…”

Section: The Products Of Open Intervals In Ordered Group Of Integersmentioning

confidence: 93%

Sums and products of intervals in ordered semigroups

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Karácsony

2021

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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We show a simple example for ordered semigroup 𝕊 = 𝕊 (+,⩽) that 𝕊 ⊆ℝ (ℝ denotes the real line) and ]a, b[ + ]c, d[ = ]a + c, b + d[ for all a, b, c, d ∈ 𝕊 such that a < b and c < d, but the intervals are no translation invariant, that is, the equation c +]a, b[ = ]c + a, c + b[ is not always fulfilled for all elements a, b, c ∈ 𝕊 such that a < b. The multiplicative version of the above example is shown too. The product of open intervals in the ordered ring of all integers (denoted by ℤ) is also investigated. Let Ix := {1, 2, . . ., x} for all x ∈ ℤ+ and defined the function g : ℤ+ → ℤ+ by g ( x ) : = max { y ∈ ℤ + | I y ⊆ I x ⋅ I x } g\left( x \right): = \max \left\{ {y \in {\mathbb{Z}_ + }|{I_y} \subseteq {I_x} \cdot {I_x}} \right\} for all x ∈ ℤ+. We give the function g implicitly using the famous Theorem of Chebishev. Finally, we formulate some questions concerning the above topics.

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Section: The Products Of Open Intervals In Ordered Group Of Integersmentioning

confidence: 93%

Sums and products of intervals in ordered semigroups

Glavosits

Karácsony

2021

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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“…For k = 3 the Bertrand's postulate was proved by Bachraoui in 2006, (see [2], Corollary 1.4. ), and for k = 4 it was proved by Loo in 2011, (see [7] ,Theorem 2.4).…”

Section: Introductionmentioning

confidence: 88%

On the Function π(x)

Bănescu

2014

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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“…This condition is automatically fulfilled for M ℓ ≥ I + 1, I = A d (N ) due to (4.3). Due to [2,Thm. 1.3] there exists at least one prime number P in the interval [2 I , 3 I ].…”

Section: Tractabilitymentioning

confidence: 99%

Multiple Lattice Rules for Multivariate $L_\infty$ Approximation in the Worst-Case Setting

Kämmerer

2019

Preprint

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We develop a general framework for estimating the L ∞ (T d ) error for the approximation of multivariate periodic functions belonging to specific reproducing kernel Hilbert spaces using approximants that are trigonometric polynomials computed from sampling values. The used sampling schemes are suitable sets of rank-1 lattices that can be constructed in an extremely efficient way. Furthermore, the structure of the sampling schemes allows for fast Fourier transform (FFT) algorithms. We present and discuss one FFT algorithm and analyze the worst case L ∞ (T d ) error for this specific approach.Using this general result we work out very weak requirements on the reproducing kernel Hilbert spaces that allow for a simple upper bound on the sampling numbers in terms of approximation numbers, where the approximation error is measured in the L ∞ (T d ) norm. Tremendous advantages of this estimate are its pre-asymptotic validity as well as its simplicity and its specification in full detail. It turns out, that approximation numbers and sampling numbers differ at most slightly. The occurring multiplicative gap does not depend on the spatial dimension d and depends at most logarithmically on the number of used linear information or sampling values, respectively.Moreover, we illustrate the capability of the new sampling method with the aid of specific highly popular source spaces, which yields that the suggested algorithm is nearly optimal from different points of view. For instance, we improve tractability results for the L ∞ (T d ) approximation for sampling methods and we achieve almost optimal sampling rates for functions of dominating mixed smoothness.Great advantages of the suggested sampling method are the constructive methods for determining sampling sets that guarantee the shown error bounds, the simplicity of all necessary implementations, i.e., the implementation of the FFT and the construction of the sampling schemes, and the extreme efficiency of all these algorithms.

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Primes in the interval [2n,3n]

Cited by 25 publications

References 0 publications

Sums and products of intervals in ordered semigroups

Sums and products of intervals in ordered semigroups

On the Function π(x)

Multiple Lattice Rules for Multivariate $L_\infty$ Approximation in the Worst-Case Setting

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