Volume 18, 2012, Number 2

This paper identifies some various methods of representing an arbitrary rhotrix. One of the methods – the row-wise method – has been chosen as it is observed to be flexible in analysing rhotrices for mathematical enrichment. A relationship between the location of the heart of a rhotrix and the dimension of the rhotrix and also a relationship between the location of the heart of a rhotrix and the order of the principal matrix of the rhotrix have been determined. The flexibility of the representation has paved way for two formulae, one for row-column multiplication of arbitrary rhotrices and the other for heart-oriented multiplication of arbitrary rhotrices. Some examples have also been given as a way of demonstrating the application of the proposed formulae. Finally, the paper introduces the concepts of subrhotrix and submatrix of a rhotrix which can be exploited for further study of various algebraic properties of rhotrices.

The characteristics of the modular ring Z₅ 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 Z₅ is complex compared with other modular rings. Even powers are restricted to three Classes for n = 4r₂ + 2 ( ̅2₄ ⊂ Z₄) but only two Classes when n = 4r₀ ( ̅0₄ ⊂ Z₄). This power distribution is also useful in the analysis of power triples.

In this article, we consider the equation x² + 19^m = yⁿ, n > 2, m > 0. We find the solutions of the title equation for not only 2 ∣ m but also 2 ∤ m.

We explicitly show that for some primes p ≡ 1 (mod 8), the elliptic curves y² = x³ − p²x and y² = x³ − 4p²x have Tate–Shafarevich groups with nontrivial elements. This involves obtaining Diophantine equations that violate the local-global principle.

We prove an open problem of Hobby and Silberger on quotients of primes in arithmetic progressions.

The numerators and denominators of the convergents of the continued fractions of 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 (c² = b² + a², 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.

Some new generalization of the Jacobsthal numbers are introduced and properties of the new number are studied.