**Volume 16** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Equations for primes obtained from integer structure**

*Original research paper. Pages 1—10*

J. V. Leyendekkers, A. Shannon

Full paper (PDF, 224 Kb) | Abstract

Integer structure illustrates how primes represented by 4*R*_{1} + 1 are equal to a sum of squares. Such primes are in Class ̅1_{4} ⊂ Z_{4}, a modular ring. The rows of squares in Z_{4} are well defined and this permits equalities for the primes to be derived from the integer structure. These equalities have the forms *R*_{1} = *r*_{1} + *r*_{0} where *r*_{1} = 3*n*(3*n* ± 1) or 2 + 9n′(n′ + 1) and *r*_{0} = 2^{2q}(12*m*(3*m* ± 1) + 1) or 2^{2q}(4(2 + 9*m*′(*m*′ + 1)) + 1), with *q *= 0, 1, 2, 3,… and *n*, *m* yielding the pentagonal numbers, and *n*′, *m*′ the triangular numbers. When *n* = *m* the equations are similar to Euler’s prime equation. Equations for the remaining primes, in Class ̅3_{4} may be obtained in the same manner using *p* = *y*^{2} − *x*^{2} with (*y* −* x*) = 1.

**Another generalization of the Fibonacci and Lucas numbers**

*Original research paper. Pages 11—17*

A. Shannon

Full paper (PDF, 136 Kb) | Abstract

This paper considers some generalizations of the Fibonacci and Lucas numbers which are essentially ratios of the former, and hence not necessarily integers. Nevertheless, some new and elegant results emerge as well as variations on well-established identities.

**Some results on infinite power towers**

*Original research paper. Pages 18—24*

Mladen Vassilev-Missana

Full paper (PDF, 223 Kb) | Abstract

In the paper the infinite power towers which are generated by an algebraic numbers belonging to the closed interval

are investigated and an answer is given to the question when they are transcendental or rational numbers. Also a necessary condition for an infinite power tower to be an irrational algebraic number is proposed.

**Note on ***φ*, *ψ* and *σ-*functions. Part 2

*Original research paper. Pages 25—28*

Krassimir Atanassov

Full paper (PDF, 125 Kb) | Abstract

An interesting property of arithmetic functions *φ*, *ψ* and *σ* is being discussed and illustrated.

**Volume 16** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4