**Volume 8** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Integer structure and constraints on powers within the modular ring Z _{4} – Part I: Even powers**

*Original research paper. Pages 41—57*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 7877 Kb) | Abstract

*n*, from equalling a power of the same value (

*n*> 2), are explored within the Modular Ring, ℤ

_{4}. Although the integer structure is complex, the results agree with those obtained for the octic or chess ring, which showed that the specific class structure, row nesting and other characteristics for powers give rise to a row exclusion factor in the modular tables of residues. This factor ensures that the resultant value of two coupled identical powers can never fit into a slot within ℤ

_{4}which is reserved for a power of the same value.

**Integer structure and constraints on powers within the modular ring Z _{4} – Part II: Odd powers**

*Original research paper. Pages 58—66*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 3398 Kb) | Abstract

_{4}. As for even powers, all pathways to an integer solution for

*c*−

^{n}*a*=

^{n}*b*,

^{n}*n*> 2, are essentially blocked by the class structure and row nesting characteristics as well as the parity requirements.

**Remark on some special numbers**

*Original research paper. Pages 67—69*

Vassia Atanassova and Krassimir Atanassov

Full paper (PDF, 1030 Kb) | Abstract

**Basic properties of weakly multiplicative functions**

*Original research paper. Pages 70—74*

Pentti Haukkanen

Full paper (PDF, 1980 Kb) | Abstract

*f*is said to be weakly multiplicative if

*f*is not identically zero and

*f*(

*np*) =

*f*(

*n*)

*f*(

*p*) for all positive integers

*n*and primes

*p*with (

*n*,

*p*) = 1. Every multiplicative function is a weakly multiplicative function but the converse is not true. In this note we study basic properties of weakly multiplicative functions with respect to the Dirichlet convolution.

**Modification of Weierstrass’s inequality**

*Original research paper. Pages 75—76*

Krassimir Atanassov

Full paper (PDF, 498 Kb)