# Volume 12, 2006, Number 3

**Volume 12** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Integer structure analysis of primes and composites from sums of two fourth powers**

*Original research paper. Pages 1—9*

J. V. Leyendekkers and A. G. Shannon

Full paper (131 Kb) | Abstract

An integer structure (IS) of the sum (*x*^{4} + *y*^{4}) is done using the modular ring Z_{6}. This sum generated many primes and the row structure of such primes is explored. The class functions of the composite factors of this sum are also given, and these, together with the associated row functions, illustrate why it is impossible to produce an integer to the fourth power from such sums. The overall results are consistent with those previously found with IS analysis.

**Using integer structure to solve Diophantine equations**

*Original research paper. Pages 10—19*

J. V. Leyendekkers and A. G. Shannon

Full paper (85 Kb) | Abstract

Diophantine equations {*ax* + *by* = *c*; *a*, *b*, *c* ∈ ℤ} are classified according to parity constraints. Various types, so classified, are solved with the theory of integer structure, via the modular ring Z_{4}. The simplest forms are those where one of the variables is confined to a single class. However, the more complex equations have solutions that follow regular (*x*, *y*) class patterns. The famous Diophantine equation in Fermat’s Last Theorem is discussed in terms of the factor structure of the sum of two powers.

**Some recurrence relations associated with the Alavi sequence**

*Original research paper. Pages 20—24*

K. T. Atanassov and A. G. Shannon

Full paper (42 Kb) | Abstract

This paper considers a modification of the Fibonacci sequence which results in the third order Alavi sequence. Not only are the initial terms quite general but the rule of formation is also modified. Some results are proved to illustrate the underlying structure of the sequence and its relation to known results in the literature. The paper concludes with a suggestion for further research with an arbitrary order extension.

**Volume 12** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4