**Volume 19** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Sum of cubes is square of sum**

*Original research paper. Pages 1–13*

Edward Barbeau and Samer Seraj

Full paper (PDF, 164 Kb) | Abstract

*n*naturals is equal to the square of their sum, we explore, for each

*n*, the Diophantine equation representing all non-trivial sets of

*n*integers with this property. We find definite answers to the standard question of infinitude of the solutions as well as several other surprising results.

**On the composition of the functions σ and φ on the set Z_{s}^{+}(P*)**

*Original research paper. Pages 14–18*

Aleksander Grytczuk

Full paper (PDF, 152 Kb) | Abstract

(*)

where σ denote the sum of divisors function and φ is the Euler’s totient function.

Let P be the set of all odd primes and

P* = {p ∈ P; p = 2

^{α}k + 1; α ≥ 1; k > 1; (k,2) = 1}.

Moreover, let

where (m

_{j}, m

_{k}) = 1; for all j ≠ k, j, k = 1, 2, …, r. In this paper we prove that if n ∈

then we have . From this and Sándor’s result it follows that (*) is true for all positive integers m ≥ 1 such that the squarefree part of m ∈ .

**Note on φ and ψ functions**

*Original research paper. Pages 19–21*

V. Kannan and R. Srikanth

Full paper (PDF, 139 Kb) | Abstract

**Note on φ, ψ and σ-functions. Part 6**

*Original research paper. Pages 22–24*

Krassimir T. Atanassov

Full paper (PDF, 118 Kb) | Abstract

*φ*(

*n*)

*ψ*(

*n*)

*σ*(

*n*) ≥

*n*

^{3}+

*n*

^{2}−

*n*− 1. connecting

*φ*,

*ψ*and

*σ*-functions is formulated and proved.

**Gaussian Jacobsthal and Gaussian Jacobsthal Lucas polynomials**

*Original research paper. Pages 25–36*

Mustafa Asci and Esref Gurel

Full paper (PDF, 164 Kb) | Abstract

*Q*matrix, determinantal representations and partial derivation of these polynomials. By defining these Gaussian polynomials for special cases

*GJ*(1) is the Gaussian Jacobsthal numbers,

_{n}*Gj*(1) is the Gaussian Jacobsthal Lucas numbers defined in {2}.

_{n}**On integer solutions of x^{4} + y^{4} – 2z^{4} – 2w^{4} = 0 **

*Original research paper. Pages 37–43*

Dustin Moody and Arman Shamsi Zargar

Full paper (PDF, 157 Kb) | Abstract

*x*

^{4}+

*y*

^{4}– 2

*z*

^{4}= 0. We find non-trivial integer solutions. Furthermore, we show that when a solution has been found, a series of other solutions can be derived. We do so using two different techniques. The first is a geometric method due to Richmond, while the second involves elliptic curves.

**New explicit formulae for the prime counting function**

*Original research paper. Pages 44–49*

Mladen Vassilev-Missana

Full paper (PDF, 159 Kb) | Abstract

**Sharp Cusa–Huygens and related inequalities**

*Original research paper. Pages 50–54*

József Sándor

Full paper (PDF, 130 Kb) | Abstract

*a*and

*b*such that

Similar sharp inequalities are also considered.

**A note on the density of the Greatest Prime Factor**

*Original research paper. Pages 55–58*

Rafael Jakimczuk

Full paper (PDF, 148 Kb) | Abstract

*P*(

*n*) be the greatest prime factor of a positive integer

*n*≥ 2. Let

*L*(

*n*) be the number of 2 ≤

*k*≤

*n*such that

*P*(

*k*) >

*k*, where 0 <

^{α}*α*< 1. We prove the following asymptotic formula

where

*ρ*(

*α*) is the Dickman’s function.

**A note on Bernoulli numbers**

*Original research paper. Pages 59–65*

Ramesh Kumar Muthumalai

Full paper (PDF, 139 Kb) | Abstract

**The structure of the Fibonacci numbers in the modular ring Z_{5}**

*Original research paper. Pages 66–72*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 149 Kb) | Abstract

_{5}.

**On new refinements of Kober’s and Jordan’s trigonometric inequalities**

*Original research paper. Pages 73–83*

József Sándor

Full paper (PDF, 174 Kb) | Abstract

*x*)/

*x*, (1 − cos

*x*)/

*x*and (tan

*x*/2)/

*x*are proved.

**Erratum to “Short remark on Jacobsthal numbers”**

*Erratum. Page 84*

Krassimir T. Atanassov

Erratum (PDF, 74 Kb)