**Volume 17** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Two modifications of Klamkin’s inequality**
*Original research paper. Pages 1—3*
Krassimir T. Atanassov

Full paper (PDF, 82 Kb) |

Abstract Two modifications of Klamkin’s inequality are formulated and proved.

**A novel approach to the discovery of ternary BBP—type formulas for polylogarithm constants**

*Original research paper. Pages 4—20*

Kunle Adegoke

Full paper (PDF, 207 Kb) | Abstract

Using clear and straightforward approaches, we prove new ternary (base 3) digit extraction BBP-type formulas for polylogarithm constants. Some known results are also rediscovered in a more direct and elegant manner. An hitherto unproved degree 4 ternary formula is also proved. Finally, a couple of ternary zero relations are established, which prove two known but hitherto unproved formulas.

**On construction of rhomtrees as graphical representation of rhotrices**

*Original research paper. Pages 21—29*

A. Mohammed and B. Sani

Full paper (PDF, 153 Kb) | Abstract

We introduce the concept of rhomtrees as a graphical method of representing rhotrices and present the relationships of their graphs with existing graphical models of some real world situations. These models include the topology of computing network, energy resource distribution network, methane compound and certain products of sets.

**A note of diagonalization of integral quadratic forms modulo ***p*^{m}

*Original research paper. Pages 30—36*

Ali H. Hakami

Full paper (PDF, 171 Kb) | Abstract

Let *m* be a positive integer, *p* be an odd prime, and ℤ_{pm} = ℤ / (*p*^{m}) be the ring of integers modulo *p*^{m}. Let *Q*(**x**) = *Q*(*x*_{1}, *x*_{2}, …, *x*_{n}) be a nonsingular quadratic form with integer coefficients. In this paper we shall prove that any nonsingular quadratic form *Q*(**x**) over ℤ, *Q*(**x**) is equivalent to a diagonal quadratic form (modulo *p*^{m}).

**Why are some right-end digits absent in primitive Pythagorean triples?**

*Original research paper. Pages 37—44*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 171 Kb) | Abstract

Integer structure analysis in the Ring Z_{3} shows that the right-end digit (RED) couples (1,4), (5,6) and (5,0) for *x*^{2}, *y*^{2} in the primitive Pythagorean triple (pPt) in the equation *z*^{2} = *x*^{2} + *y*^{2} do not lead to the primitive form of triple. The rows of *x*^{2}, *y*^{2} with these REDs do not add to the required form for *z*^{2}. Since 3 ∤ *z* , the row of *z*^{2} must follow the pentagonal numbers. Common factors for *x*, *y* are also inconsistent with pPt formation so that the (*x*^{2}, *y*^{2}). RED (5,0) may be discarded directly.

**Volume 17** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4