**Volume 6** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**The Goldberg-conjecture primes within a modular ring**

*Original research paper. Pages 101–112*

J. Leyendekkers and A. Shannon

Full paper (PDF, 482 Kb) | Abstract

_{4 }in relation to Goldbach’s Conjecture. Such analyses, together with the identification of compatible right-end digits for the Goldberg ’system’, permit a more efficient search for prime pairs. This is useful for the study of very large even numbers, the distribution of twin primes and other prime constellations, and the relative distribution of primes between the classes l

_{4 }and 3

_{4. }

**Analysis of the roots of some Cardano cubes**

*Original research paper. Pages 113–117*

J. Leyendekkers and A. Shannon

Full paper (PDF, 175 Kb) | Abstract

$$ e = \frac{2q^2R \tan^2 \theta}{3 – \tan^2 \theta}$$

$$ E = 2 \Big \{ \Big ( \frac{3}{3 – \tan^2 \theta} \Big )^{1 \over 2} – 1 \Big \}$$

with $R = \frac{p}{q} = h(\theta)$ and 11° < $\theta$ < 60° for real zero. Furthermore, for $E$ integer, the range of $\theta$ is reduced to 52° < $\theta$ < 60°, where the functional surfaces suggest the reason the integer $E$ would only be compatible with an irrational value of $R$. This is verified algebraically. [/expand]

**Expressions for the Dirichlet inverse of arithmetical functions**

*Original research paper. Pages 118–124*

P. Haukkanen

Full paper (PDF, 222 Kb) | AbstractWe express the values of the Dirichlet inverse

*f*in terms of the values of

^{ -1}*f*without using the values of

*f*. We use a method based on representing

^{ -1}*f**

^{ -1}*f*= δ as a system of linear equations. Jagannathan has given many of the results of this paper without proof starting from the basic recurrence relation for the values of

*f*.

^{ -1}**Note on cyclotomic polynomials and Legendre symbol**

*Original research paper. Pages 125–128*

M. Vassilev-Missana

Full paper (PDF, 134 Kb)

**Short remark on number theory. II**

*Original research paper. Pages 129–130*

K. Atanassov

Full paper (PDF, 73 Kb)