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On the Integrality of the Elementary Symmetric Functions of 1, 1/3, . . . , 1/(2n − 1)
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Chunlin Wang
Published/Copyright:
December 9, 2015
Abstract
Erdős and Niven proved that for any positive integers m and d, there are only finitely many positive integers n for which one or more of the elementary symmetric functions of 1/m, 1/(m + d), . . . , 1/(m + nd) are integers. In this paper, we show that if n ≥ 2, then none of the elementary symmetric functions of 1, 1/3, . . . , 1/(2n − 1) is an integer
Keywords: elementary symmetric function; harmonic series
References
[1] DUSART, P.: Estimates of some functions over primes without R. H., arXiv:1002.0442 (To appear).Search in Google Scholar
[2] ERDŐS, P.-NIVEN, I.: Some properties of partial sums of the harmonic series, Bull. Amer. Math. Soc. 52 (1946), 248-251.10.1090/S0002-9904-1946-08550-XSearch in Google Scholar
[3] SCHOENFELD, L.: Sharper bounds for the Chebyshev functions θ(x) and ψ(x), II, Math. Comp. 30 (1976), 337-360.Search in Google Scholar
Received: 2012-11-13
Accepted: 2013-1-30
Published Online: 2015-12-9
Published in Print: 2015-10-1
Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Finite Mixed Sums wih Harmonic Terms
- Packing of ℝ2 by Crosses
- On the Integrality of the Elementary Symmetric Functions of 1, 1/3, . . . , 1/(2n − 1)
- Generalized Derivations as a Generalization of Jordan Homomorphisms Acting on Lie Ideals and Right Ideals
- Generalized Derivations on Lie Ideals and Power Values on Prime Rings
- On Monoids of Injective Partial Cofinite Selfmaps
- Extensions of Dynamic Inequalities of Hardy’s Type on Time Scales
- The Controlled Convergence Theorem for the Gap-Integral
- The Solvability of a Nonlocal Boundary Value Problem
- Oscillation Criteria for Third Order Differential Equations with Functional Arguments
- Asymptotic Behavior of Solutions of a Nonlinear Neutral Generalized Pantograph Equation with Impulses
- On Null Lagrangians
- Principal Eigenvalues for Systems of Schrödinger Equations Defined in the whole Space with Indefinite Weights
- Convergence of Series on Large Set of Indices
- On Approximation Properties of a New Type of Bernstein-Durrmeyer Operators
- Representation of Extendible Bilinear Forms
- Spectra and Fine Spectra of Lower Triangular Double-Band Matrices as Operators on Lp (1 ≤ p < ∞)
- Topological Fundamental Groups and Small Generated Coverings
- A Relation between two Kinds of Norms for Martingales
- Linearization Regions in Singular Weakly Nonlinear Regression Models with Constraints
- Parametric Equilibrium Problems Governed by Topologically Pseudomonotone Bifunctions
- Identification of a Parameter in Fourth-Order Partial Differential Equations by an Equation Error Approach
