**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

*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*, it turns out that the group of rational points on

_{q}*E*is cyclic of order

*q*+1. This group is a product of cyclic groups if

*R*=

*Z*, the ring of integers modulo a square-free

_{n}*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*and

_{q}*Z*.

_{n}**Some recurrence relations for binary sequence matrices**

*Original research paper. Pages 9–13*

A. G. Shannon

Full paper (PDF, 34 Kb) | Abstract

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

*Original research paper. Pages 14–17*

Yilun Shang

Full paper (PDF, 166 Kb) | Abstract

**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

*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

**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

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

*Original research paper. Pages 42–49*

Yasutsugu Fujita and Alain Togbé

Full paper (PDF, 194 Kb) | Abstract

*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

**The structure of ‘Pi’**

*Original research paper. Pages 61–68*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 150 Kb) | Abstract

_{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

*φ*and

*ψ*-functions are formulated and proved.