**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

This paper is concerned with integer solutions of the Diophantine equation *x*_{1}^{4} +* x*_{2}^{4} + *x*_{3}^{4} = *k x*_{4}^{2} where * 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}^{2} where * 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

In this paper we consider Diophantine triples {*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

Only prime integers that are in Class ̅1_{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

Two triangular number based primality tests for numbers in the arithmetic progressions 8*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

We revisit Elder’s theorem on integer partitions, which is a generalization of Stanley’s theorem. Two new proofs are presented. The first proof is based on certain tilings of 1 × ∞ boards while the second one is a consequence of a more general identity we prove using generating functions.

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

*Original research paper. Pages 36—39*

Simon Davis

Full paper (PDF, 150 Kb) | Abstract

The existence of prime values of meromorphic functions and irreducible polynomials is considered. An argument for an infinite number of prime values of irreducible polynomials over

is provided.

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

*Original research paper. Pages 40—47*

József Sándor and Edith Egri

Full paper (PDF, 168 Kb) | Abstract

We consider certain properties of functions *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

Consider the multiplicative group of integers modulo *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

The structure of the ‘Golden Ratio Family’ is consistent enough to permit the primality tests developed for *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

We consider *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 generalization of Prof. Ivan Prodanov’s inequality is formulated and proved. As a partial case, an inequality discussed in D. Mitrinovic and M. Popadic’s book

is obtained.

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

*Original research paper. Pages 74—80*

Mark Shattuck

Full paper (PDF, 157 Kb) | Abstract

In this note, we provide a combinatorial proof of a generalized recurrence formula satisfied by the Stirling numbers of the second kind. We obtain two extensions of this formula, one in terms of *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

* *

* *

* *

* *