**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 o f compatible right-end digits for the Goldberg ’system’, permit a more efficient search for prime pairs. This is useful for the study o f very large even numbers, the distribution o f twin primes and other prime constellations, and the relative distribution o f 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

**#!**The Cardano cubic, y3 — 6pqy — 3pq(p + q), p, q € Z+, has one real zero and a complex conjugate pair. The real zero is given by 2(2pq + e)5 or (E 4- 2){2pq)^, in which e, E are important parameters that feature in the roots of all Cardano cubics. They are functions of the coefficients of the complex conjugate pairs. We find that 2 q2 R tan2 9 3 — tan2 9 ’ i e = with R = P- = h(9) and 11° < 9 < 60° for real zero. Futhermore, for E integer, the range of q 9 is reduced to 52° < 9 < 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.

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

*Original research paper. Pages 118—124*

P. Haukkanen

Full paper (PDF, 222 Kb) | Abstract

*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)