Volume 4 ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4
The characteristics of primes and other integers within the modular ring Z4 and in Class ![Rendered by QuickLaTeX.com \overline{1}](https://nntdm.net/wp-content/ql-cache/quicklatex.com-69e3e007a690a9136f3cb1c7755f3df5_l3.png)
Original research paper. Pages 1–17
J. V. Leyendekkers, J. M. Rybak and A. G. Shannon
Full paper (PDF, 663 Kb) | Abstract
The integer structure of Class
![Rendered by QuickLaTeX.com \overline{1}](https://nntdm.net/wp-content/ql-cache/quicklatex.com-69e3e007a690a9136f3cb1c7755f3df5_l3.png)
in the modular ring
Z4 has been analysed in detail. Most integers of this category equal a sum of two squares (
![Rendered by QuickLaTeX.com x^2 + y^2](https://nntdm.net/wp-content/ql-cache/quicklatex.com-f68a3dbbb12efc4f707bac0bced080c8_l3.png)
). Those that do not are non-primes. The primes are distinguished by having a unique
![Rendered by QuickLaTeX.com \langle x, y \rangle](https://nntdm.net/wp-content/ql-cache/quicklatex.com-cb9277a55020022e973a333737685eaf_l3.png)
pair that has no common factors. Other integers in Class
![Rendered by QuickLaTeX.com \overline{1}](https://nntdm.net/wp-content/ql-cache/quicklatex.com-69e3e007a690a9136f3cb1c7755f3df5_l3.png)
have multiple values of
![Rendered by QuickLaTeX.com \langle x, y \rangle](https://nntdm.net/wp-content/ql-cache/quicklatex.com-cb9277a55020022e973a333737685eaf_l3.png)
or more rarely a single
![Rendered by QuickLaTeX.com \langle x, y \rangle](https://nntdm.net/wp-content/ql-cache/quicklatex.com-cb9277a55020022e973a333737685eaf_l3.png)
pair with common factors. Methods of estimating
![Rendered by QuickLaTeX.com \langle x, y \rangle](https://nntdm.net/wp-content/ql-cache/quicklatex.com-cb9277a55020022e973a333737685eaf_l3.png)
pairs are given. These are based on the class structure within
Z4 and the right-most end digit characteristics. The identification of primes is consequently facilitated.
The characteristics of primes and other integers within the modular ring Z4 and in class ![Rendered by QuickLaTeX.com \overline{3}](https://nntdm.net/wp-content/ql-cache/quicklatex.com-0fd9b21f56e1fc02921b031253f688d2_l3.png)
Original research paper. Pages 18–37
J. V. Leyendekkers, J. M. Rybak and A. G. Shannon
Full paper (PDF, 911 Kb) | Abstract
Integers,
n, in Class
![Rendered by QuickLaTeX.com \overline{3}](https://nntdm.net/wp-content/ql-cache/quicklatex.com-0fd9b21f56e1fc02921b031253f688d2_l3.png)
of the modular ring
Z4 have been analysed in detail. All
n equal the difference of squares,
![Rendered by QuickLaTeX.com x^2 - y^2](https://nntdm.net/wp-content/ql-cache/quicklatex.com-073cd85fd4c598419064161b4b6aeda2_l3.png)
, with
![Rendered by QuickLaTeX.com x](https://nntdm.net/wp-content/ql-cache/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png)
even and
![Rendered by QuickLaTeX.com y](https://nntdm.net/wp-content/ql-cache/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png)
odd. Primes are distinguished by having only one
![Rendered by QuickLaTeX.com \langle x, y \rangle](https://nntdm.net/wp-content/ql-cache/quicklatex.com-cb9277a55020022e973a333737685eaf_l3.png)
pair:
![Rendered by QuickLaTeX.com \langle X, Y \rangle](https://nntdm.net/wp-content/ql-cache/quicklatex.com-99d576240b65a34a5d4e0cb43c215391_l3.png)
with
![Rendered by QuickLaTeX.com X - Y = 1](https://nntdm.net/wp-content/ql-cache/quicklatex.com-fe46335b8d0ed55efe7f0e5c4c50e27e_l3.png)
, and
![Rendered by QuickLaTeX.com X = (n + 1) / 2](https://nntdm.net/wp-content/ql-cache/quicklatex.com-06e5b2e2a59040e8cfcb3f29187a4143_l3.png)
. In this paper four methods of calculating the
![Rendered by QuickLaTeX.com \langle x, y \rangle](https://nntdm.net/wp-content/ql-cache/quicklatex.com-cb9277a55020022e973a333737685eaf_l3.png)
values are given. These methods are based on prime factorisation, the
Z4 class structure, and Fermat’s Little Theorem. Mersenne primes are uniquely distributed within Class
![Rendered by QuickLaTeX.com \overline{3}](https://nntdm.net/wp-content/ql-cache/quicklatex.com-0fd9b21f56e1fc02921b031253f688d2_l3.png)
and some new features of these primes are also stated.
Numerical properties of Morgan-Voyce Numbers
Original research paper. Pages 38–42
A. F. Horadam
Full paper (PDF, 163 Kb)
One extremal problem. 8
Original research paper. Pages 43–44
Krassimir T. Atanassov
Full paper (PDF, 131 Kb)
Volume 4 ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4