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Five Curious Congruences Modulo p2

  • Romeo Meštrović EMAIL logo
Veröffentlicht/Copyright: 29. Juli 2015
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Mathematica Slovaca
Aus der Zeitschrift Mathematica Slovaca Band 65 Heft 3

Abstract

Let p ≥ 3 be a prime, and let qp(2) := (2P 1 - 1)/p be the Fermat quotient of p to base 2. In this note, we prove that

.

As an application, using two combinatorial identities due to T. B. Staver in 1947 and G. Galperin and H. Gauchman in 2004, we obtain four curious combinatorial congruences modulo p2. As an auxiliary result, here we present an elementary proof of a congruence established by E. Lehmer in 1938. Notice that this con­gruence together with some other congruences given in our lemmas leads to the elementary proof of a beautiful Morley’s congruence due to F. Morley in 1895.

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Received: 2012-8-26
Accepted: 2012-10-8
Published Online: 2015-7-29
Published in Print: 2015-6-1

© Mathematical Institute Slovak Academy of Sciences

Artikel in diesem Heft

  1. Fuzzy Pseudo Subalgebras and Ideals of Pseudo D-Algebras
  2. Five Curious Congruences Modulo p2
  3. On the Hybrid Mean Value Involving Dedekind Sums and Kloosterman Sums
  4. Constructing New Crossed Group Categories Over Weak Hopf Group Algebras
  5. Generalization of Hilbert Inequality with Some Parameters
  6. Some Applications of Differential Subordination of p-Valent Functions
  7. Coefficient Estimates and Quasi-Subordination Properties Associated with Certain Subclasses of Analytic and Bi-Univalent Functions
  8. On Properties of Entire Solutions of Difference Equations and Difference Polynomials
  9. On the Ψ-Strong Stability of Nonlinear Lyapunov Matrix Differential Equations
  10. On the Existence of Solutions of Ordinary Differential Equations in Banach Spaces
  11. Global Dynamics of a Delayed n + m-Species Competition Predator-Prey System on Time Scales
  12. Comparison Of (G′/G)-Methods for Finding Exact Solutions of the Drinfeld-Sokolov System
  13. A Generalization of Cyclic Amenability of Banach Algebras
  14. n-Weak Module Amenability of Triangular Banach Algebras
  15. On a Class of Operator-Differential Equations of the Third Order With Multiple Characteristics on the Whole Axis in the Weighted Space
  16. Optimality Conditions for Vector Equilibrium Problems with Set-Valued Mappings and Cone Constraints
  17. Classification of the Blaschke Isoparametric Hypersurfaces in Lorentzian Space Forms
  18. A New Proof of a Theorem on Long Cycles
https://doi.org/10.1515/ms-2015-0033

Artikel in diesem Heft

  1. Fuzzy Pseudo Subalgebras and Ideals of Pseudo D-Algebras
  2. Five Curious Congruences Modulo p2
  3. On the Hybrid Mean Value Involving Dedekind Sums and Kloosterman Sums
  4. Constructing New Crossed Group Categories Over Weak Hopf Group Algebras
  5. Generalization of Hilbert Inequality with Some Parameters
  6. Some Applications of Differential Subordination of p-Valent Functions
  7. Coefficient Estimates and Quasi-Subordination Properties Associated with Certain Subclasses of Analytic and Bi-Univalent Functions
  8. On Properties of Entire Solutions of Difference Equations and Difference Polynomials
  9. On the Ψ-Strong Stability of Nonlinear Lyapunov Matrix Differential Equations
  10. On the Existence of Solutions of Ordinary Differential Equations in Banach Spaces
  11. Global Dynamics of a Delayed n + m-Species Competition Predator-Prey System on Time Scales
  12. Comparison Of (G′/G)-Methods for Finding Exact Solutions of the Drinfeld-Sokolov System
  13. A Generalization of Cyclic Amenability of Banach Algebras
  14. n-Weak Module Amenability of Triangular Banach Algebras
  15. On a Class of Operator-Differential Equations of the Third Order With Multiple Characteristics on the Whole Axis in the Weighted Space
  16. Optimality Conditions for Vector Equilibrium Problems with Set-Valued Mappings and Cone Constraints
  17. Classification of the Blaschke Isoparametric Hypersurfaces in Lorentzian Space Forms
  18. A New Proof of a Theorem on Long Cycles
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