**Volume 21** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Some Diophantine equations concerning biquadrates**

*Original research paper. Pages 1–5*

Ajai Choudhry

Full paper (PDF, 130 Kb) | Abstract

*x*

_{1}

*+*

^{4}*x*

_{2}

*+*

^{4}*x*

_{3}

*=*

^{4}*k x*

_{4}

*where*

^{2}*k*is a given positive integer. Till now, integer and parametric solutions of this Diophantine equation have been published only when

*k*= 1 or 2 or 3. In this paper we obtain parametric solutions of this equation for 43 values of

*k*≤ 100. We also show that the equation cannot have any solution in integers for 54 values of

*k*≤ 100. The solvability of the equation

*x*

_{1}

*+*

^{4}*x*

_{2}

*+*

^{4}*x*

_{3}

*=*

^{4}*k x*

_{4}

*where*

^{2}*k*could not be determined for three values of k ≤ 100, namely 34, 35 and 65.

**On Diophantine triples and quadruples**

*Original research paper. Pages 6–16*

Yifan Zhang and G. Grossman

Full paper (PDF, 196 Kb) | Abstract

*a, b, c*} (denoted

*D*(

*n*)-3-tuples) and give necessary and sufficient conditions for existence of integer

*n*given the 3-tuple {

*a, b, c*} so that

*ab + n, ac + n, bc + n*are all squares of integers. Several examples as applications of the main results, related to both Diophantine triples and quadruples, are given.

**The sum of squares for primes**

*Original research paper. Pages 17–21*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 87 Kb) | Abstract

_{4}of the Modular Ring Z

_{4}equate to a sum of squares of integers

*x*and

*y*. A simple equation to predict these integers is developed which distinguishes prime and composite numbers in that one

*(x*,

*y*) couple exists for primes, but composites have either one couple with a common factor or the same number of couples as there are factors. In particular, composite Fibonacci numbers always have multiple

*(x*,

*y*) couples because the factors are all elements of ̅1

_{4}.

**Two triangular number primality tests and twin prime counting in arithmetic progressions of modulus 8**

*Original research paper. Pages 22–29*

Werner Hürlimann

Full paper (PDF, 103 Kb) | Abstract

*n*± 1 are obtained. Their use yield a new Diophantine approach to the existence of an infinite number of twin primes of the form (8

*n*−1, 8

*n*+1).

**Generating function and combinatorial proofs of Elder’s theorem**

*Original research paper. Pages 30–35*

Robson da Silva, Jorge F. A. Lima, José Plínio O. Santos and Eduardo C. Stabel

Full paper (PDF, 208 Kb) | Abstract

**Prime values of meromorphic functions and irreducible polynomials**

*Original research paper. Pages 36–39*

Simon Davis

Full paper (PDF, 150 Kb) | Abstract

**On ( M, N)-convex functions**

*Original research paper. Pages 40–47*

József Sándor and Edith Egri

Full paper (PDF, 168 Kb) | Abstract

*f*:

*J*→

*I*(

*I*,

*J*intervals) such that

*f*(

*M*(

*x*,

*y*)) ≤

*N*(

*f*(

*x*),

*f*(

*y*)), where

*M*and

*N*are general means. Some results are extensions of the case

*M = N = L*, where

*L*is the logarithmic mean.

**On the number of semi-primitive roots modulo n**

*Original research paper. Pages 48–55*

Pinkimani Goswami and Madan Mohan Singh

Full paper (PDF, 176 Kb) | Abstract

*n*, denoted by ℤ*

_{n}. An element

*a*∈ ℤ*

_{n}is said to be a semi-primitive root modulo

*n*if the order of

*a*is φ (

*n*)/2, where φ(

*n*) is the Euler’s phi-function. In this paper, we’ll discuss on the number of semi-primitive roots of non-cyclic group ℤ*

_{n}and study the relation between

*S*(

*n*) and

*K*(

*n*), where

*S*(

*n*) is the set of all semi-primitive roots of non-cyclic group ℤ*

_{n}and

*K*(

*n*) is the set of all quadratic non-residues modulo

*n*.

**Primes within generalized Fibonacci sequences**

*Original research paper. Pages 56–63*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 117 Kb) | Abstract

*φ*

_{5}to be applicable. Moreover, the factors of the composite numbers formed by a prime subscripted member of the sequence adhere to the same pattern as for

*φ*

_{5}. Only restricted modular class structures allow prime subscripted members of the sequence to be a sum of squares. Furthermore, other properties of

*φ*

_{5}are found to apply to those other members with structural compatibility.

**On some q-Pascal’s like triangles**

*Original research paper. Pages 64–69*

Toufik Mansour and Matthias Schork

Full paper (PDF, 192 Kb) | Abstract

*q*-analogs of Pascal’s like triangles, which were studied by Atanassov in a series of papers.

**A generalization of Ivan Prodanov’s inequality**

*Original research paper. Pages 70–73*

Krassimir T. Atanassov

Full paper (PDF, 121 Kb) | Abstract

**A generalized recurrence formula for Stirling numbers and related sequences**

*Original research paper. Pages 74–80*

Mark Shattuck

Full paper (PDF, 157 Kb) | Abstract

*r*-Whitney numbers and another in terms of

*q*-Stirling numbers of Carlitz. Modifying our proof yields analogous formulas satisfied by the r-Stirling numbers of the first kind and by the

*r*-Lah numbers.

**Volume 21** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

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