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

**Permutation polynomials and elliptic curves**
*Original research paper. Pages 1—8*
Yotsanan Meemark and Attawut Wongpradit

Full paper (PDF, 188 Kb) |

Abstract In this work, we study the elliptic curve *E* : *y*^{2} = *f*(*x*), where *f*(*x*) is a cubic permutation polynomial over some finite commutative ring *R*. In case *R* is the finite field *F*_{q}, it turns out that the group of rational points on *E* is cyclic of order *q* +1. This group is a product of cyclic groups if *R* = *Z*_{n}, the ring of integers modulo a square-free *n*. In addition, we introduce a shift-invariant elliptic curve which is an elliptic curve *E* : *y*^{2} = *f*(*x*), where *y*^{2} − *f*(*x*) is a weak permutation polynomial. We end our paper with a necessary and sufficient condition for the existence of a shift-invariant elliptic curve over *F*_{q} and *Z*_{n}.

**Some recurrence relations for binary sequence matrices**

*Original research paper. Pages 9—13*

A. G. Shannon

Full paper (PDF, 34 Kb) | Abstract

This note compares and contrasts some properties of binary sequences with matrices and associated recurrence relations in order to stimulate some enrichment exercises and pattern puzzles.

**Degree sequence of configuration model with vertex faults**

*Original research paper. Pages 14—17*

Yilun Shang

Full paper (PDF, 166 Kb) | Abstract

We study the degree sequence of configuration model of random graphs with random vertex deletion. The degree sequences are characterized under various deletion probabilities. Our results have implications in communication networks where random faults due to inner consumption and outer disturbance often occur.

**A class of digit extraction BBP-type formulas in general binary bases**

*Original research paper. Pages 18—32*

Kunle Adegoke, Jaume Oliver Lafont and Olawanle Layeni

Full paper (PDF, 206 Kb) | Abstract

BBP-type formulas are usually discovered experimentally, one at a time and in specific bases, through computer searches. In this paper, however, we derive explicit digit extraction BBPtype formulas in general binary bases *b* = 2^{12p}, for *p* ∈ Z^{+} and mod (*p*, 2) = 1. As particular examples, new binary formulas are presented for *π*√3, *π*√3log2, √3Cl_{2}(*π*/3) and a couple of other polylogaritm constants. A variant of the formula for *π*√3log2 derived in this paper has been known for over ten years but was hitherto unproved. Binary BBP-type formulas for the logarithms of an infinite set of primes and binary BBP-type representations for the arctangents of an infinite set of rational numbers are also presented. Finally, new binary BBP-type zero relations are established.

**Solution to an open problem by Rooin**

*Original research paper. Pages 33—36*

V. Lokesha, K. M. Nagaraja, Naveen Kumar B. and Sandeep Kumar

Full paper (PDF, 154 Kb) | Abstract

In this note, we obtained the solution to an open problem posed by J. Rooin, using Levinson’s Inequality.

**Schur convexity of Gnan mean for two variables**

*Original research paper. Pages 37—41*

V. Lokesha, K. M. Nagaraja, Naveen Kumar B. and Y.-D. Wu

Full paper (PDF, 185 Kb) | Abstract

In this paper, the convexity and Schur convexity of the Gnan mean and its dual form in two variables are discussed.

**Uniqueness of the extension of the ***D*(4*k*^{2})-triple {*k*^{2} – 4, *k*^{2}, 4*k*^{2} – 4}

*Original research paper. Pages 42—49*

Yasutsugu Fujita and Alain Togbé

Full paper (PDF, 194 Kb) | Abstract

Let n be a nonzero integer. A set of *m* distinct positive integers is called a *D*(*n*)-*m*-tuple if the product of any two of them increased by *n* is a perfect square. Let *k* be an integer greater than two. In this paper, we show that if {*k*^{2} − 4, *k*^{2}, 4*k*^{2} − 4, *d*} is a D(4*k*^{2})-quadruple, then *d* = 4*k*^{4} − 8*k*^{2}.

**On some identities of Ramanujan—Göllnitz—Gordon continued fraction**

*Original research paper. Pages 50—60*

K. R. Vasuki, G. Sharath and K. R. Rajanna

Full paper (PDF, 174 Kb) | Abstract

In this paper, we give an alternative and simple proof of certain identities of Ramanujan–Göllnitz–Gordon continued fraction.

**The structure of ‘Pi’**

*Original research paper. Pages 61—68*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 150 Kb) | Abstract

Classes of the modular ring Z_{4} were substituted into convergent infinite series for *π* and √2 to obtain *Q*, the ratio of the arc of a circle to the side of an inscribed square to yield *π* = 2√2 *Q*. The corresponding convergents of the continued fractions for π, √2 and *Q* were then considered, together with the class patterns of the modular rings {Z_{4}, Z_{5}, Z_{6}} and decimal patterns for *π*.

**Note on ***φ*, *ψ* and *σ*-functions. Part 4

*Original research paper. Pages 69—72*

Krassimir T. Atanassov

Full paper (PDF, 123 Kb) | Abstract

Two inequalities connecting *φ* and *ψ*-functions are formulated and proved.

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