**Volume 23** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**A note on the quartic Diophantine equation A^{4} + hB^{4} = C^{4} + hD^{4}**

*Original research paper. Pages 1—3*

Ajai Choudhry

Full paper (PDF, 113 Kb) | Abstract

*A*

^{4}+

*hB*

^{4}=

*C*

^{4}+

*hD*

^{4}are known for all positive integer values of

*h*< 1000. While a solution of the aforementioned diophantine equation for any arbitrary positive integer value of

*h*is not known, Gerardin and Piezas found solutions of this equation when

*h*is given by polynomials of degrees 5 and 2, respectively. In this paper, we present several new solutions of this equation when

*h*is given by polynomials of degrees 2, 3 and 4.

**Refinements of the Mitrinović–Adamović inequality, with application**

*Original research paper. Pages 4—6*

József Sándor

Full paper (PDF, 139 Kb) | Abstract

**New theorems on explicit evaluation of a parameter of Ramanujan’s χ(q) function**

*Original research paper. Pages 7—18*

Nipen Saikia and Jubaraj Chetry

Full paper (PDF, 184 Kb) | Abstract

*I*, for positive real number

_{k,n}*n*and

*k*, involving Ramanujan’s

*χ*(

*q*) function by establishing its connection with some other parameters of Ramanujan’s theta-functions. As applications of the parameter

*I*we offer formulas for the explicit values of Ramanujan’s cubic continued fraction and

_{k,n}*χ*(e

^{−πn}).

**Inequalities for φ and ψ functions (III)**

*Original research paper. Pages 19—23*

Krassimir T. Atanassov

Full paper (PDF, 148 Kb) | Abstract

**Ascending sequences with neighboring elements add up to perfect square numbers**

*Original research paper. Pages 24—27*

Kai Jin

Full paper (PDF, 128 Kb) | Abstract

*n*to ascending sequences as few as possible, so that every neighboring pair of elements in each sequence add up to some perfect square number. We prove that the minimum number of sequences is . We hope that this paper exhibits an interesting property of the perfect square numbers.

**Some enumerations of non-trivial composition of the differential operations and the directional derivative**

*Original research paper. Pages 28—38*

Ivana Jovović and Branko Malešević

Full paper (PDF, 179 Kb) | Abstract

^{n}(

*n*≥ 3). One new enumeration of the higher order non-trivial compositions is obtained.

**Further results on arctangent sums with applications to generalized Fibonacci numbers**

*Original research paper. Pages 39—53*

Robert Frontczak

Full paper (PDF, 206 Kb) | Abstract

**Primitive Pythagorean triples and generalized Fibonacci sequences**

*Original research paper. Pages 54—62*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 206 Kb) | Abstract

*r*

_{1}+ 1) where

*r*

_{1}belongs to the class in the modular ring

*Z*

_{4}.

**Some combinatorial formulas for the partial r-Bell polynomials**

*Original research paper. Pages 63—76*

Mark Shattuck

Full paper (PDF, 208 Kb) | Abstract

*r*-Bell polynomials generalize the classical partial Bell polynomials (coinciding with them when

*r*= 0) by assigning a possibly different set of weights to the blocks containing the

*r*smallest elements of a partition no two of which are allowed to belong to the same block. In this paper, we study the partial

*r*-Bell polynomials from a combinatorial standpoint and derive several new formulas. We prove some general identities valid for arbitrary values of the parameters as well as establish formulas for some specific evaluations. Several of our results extend known formulas for the partial Bell polynomials and reduce to them when

*r*= 0. Our arguments are largely combinatorial, and therefore provide, alternatively, bijective proofs of these formulas, many of which were shown by algebraic methods.

**An elementary alternative proof for Chan’s analogue of Ramanujan’s most beautiful identity and some inequality of the cubic partition**

*Original research paper. Pages 77—87*

Koustav Banerjee and Prabir Das Adhikary

Full paper (PDF, 193 Kb) | Abstract

*a*(

*n*). That apart we also examine inequalities

*a*(

*n*) and provide upper bound for it in the fashion of the classic partition function

*p*(

*n*).

**A Fibonacci integral lattice approach to Pythagoras’ Theorem**

*Original research paper. Pages 88—90*

Anthony G. Shannon and John N. Crothers

Full paper (PDF, 55 Kb) | Abstract

*a*,

*b*), (

*b*, –

*a*)}, where

*a*and

*b*are successive Fibonacci numbers, are employed to develop intermediate convergence forms of Pythagoras’ Theorem for triangles with integral sides.

**On the bounds for the norms of circulant matrices with the Jacobsthal and Jacobsthal–Lucas numbers**

*Original research paper. Pages 91—98*

Ş. Uygun and S. Yaşamalı

Full paper (PDF, 175 Kb) | Abstract

*A*=

*C*(

_{r}*j*

_{0},

*j*

_{1}, …,

*j*

_{n−1}) and

*B*=

*C*(

_{r}*c*

_{0},

*c*

_{1}, …,

*c*

_{n−1}).

**A note on the density of quotients of primes in arithmetic progressions**

*Original research paper. Pages 99—100*

Brian D. Sittinger

Full paper (PDF, 133 Kb) | Abstract

**On some cancellation algorithms**

*Original research paper. Pages 101—114*

Andrzej Tomski and Maciej Zakarczemny

Full paper (PDF, 224 Kb) | Abstract

*f*be a natural-valued function defined on the Cartesian product of finitely many copies of ℕ (positive integers). Here we will discuss some modifications of the sieve of Eratosthenes in the sense that we cancel the divisors of all possible values of

*f*in the points whose sum of coordinates is less or equal to

*n*. By applying similar arguments to those used in the paper (J. Browkin, H-Q. Cao,

*Modifications of the Eratosthenes sieve*, Colloq. Math. 135, (2014)), but also in the companion papers, we investigate new problems for the values of some polynomial functions or quadratic and cubic forms.