**Volume 18** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Row-wise representation of arbitrary rhotrix**

*Original research paper. Pages 1—27*

Chinedu M. Peter

Full paper (PDF, 662 Kb) | Abstract

**The modular ring Z_{5}**

*Original research paper. Pages 28—33*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 46 Kb) | Abstract

_{5}are discussed. Each of the five classes is specific for two right-end-digits (REDs), even and odd. This facilitates analysis of the primitive Pythagorean triples, namely, the factors of components and the structure that prevents the two minor components from having REDs of (1,4), (5,6), (5,0) or (9,0). The RED feature is also useful in solving quadratic equations and the quick identification of modular classes. The distribution of powers within Z

_{5}is complex compared with other modular rings. Even powers are restricted to three Classes for

*n*= 4

*r*

_{2}+ 2 ( ̅2

_{4}⊂ Z

_{4}) but only two Classes when

*n*= 4

*r*

_{0}( ̅0

_{4}⊂ Z

_{4}). This power distribution is also useful in the analysis of power triples.

**On the solutions of the equation x^{2} + 19^{m} = y^{n} **

*Original research paper. Pages 34—41*

Bilge Peker and Selin (İnağ) Çenberci

Full paper (PDF, 208 Kb) | Abstract

*x*

^{2}+ 19

*=*

^{m}*y*,

^{n}*n*> 2,

*m*> 0. We find the solutions of the title equation for not only 2 ∣

*m*but also 2 ∤

*m*.

**On Tate—Shafarevich groups of families of elliptic curves**

*Original research paper. Pages 42—55*

Jerome T. Dimabayao and Fidel R. Nemenzo

Full paper (PDF, 228 Kb) | Abstract

*p*≡ 1 (mod 8), the elliptic curves

*y*

^{2}=

*x*

^{3}−

*p*

^{2}

*x*and

*y*

^{2}=

*x*

^{3}− 4

*p*

^{2}

*x*have Tate—Shafarevich groups with nontrivial elements. This involves obtaining Diophantine equations that violate the local-global principle.

**Quotients of primes in arithmetic progressions**

*Original research paper. Pages 56—57*

Ace Micholson

Full paper (PDF, 64 Kb) | Abstract

**Pellian sequence relationships among π, e, √2**

*Original research paper. Pages 58—62*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 143 Kb) | Abstract

*π*,

*e*and √2 are shown to be elements of second order recurrence sequences of the Pellian or Fibonacci variety which are related to Pythagorean triples (

*c*

^{2}=

*b*

^{2}+

*a*

^{2},

*b*>

*a*).

*π*and √2 have surprisingly similar structures except that √2 has primitive Pythagorean triples with

*c − b*= 1 or

*b − a*= 1, whereas π has

*c − b*even and not constant and

*b − a*not constant, although the right-end-digits are constant.

**Short remarks on Jacobsthal numbers**

*Original research paper. Pages 63—64*

Krassimir Atanassov

Full paper (PDF, 104 Kb) * * | Erratum (PDF, 74 Kb) | Abstract