**Volume 10** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**The row structure of squares in modular rings**

*Original research paper. Pages 1—11*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 159 Kb) | Abstract

_{6}are analysed. This structure is shown to be linked via generalized pentagonal numbers to that of the odd squares. When 3 |

*N*, the link is via the triangular numbers. Equations could thus be developed for the rows of those primes that equal a sum of squares. Since the results are general they can be used to study in some depth those systems that have squares as a dominant feature.

**The structure of Fibonacci numbers in modular rings**

*Original research paper. Pages 12—23*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 130 Kb) | Abstract

_{4}. It is found that the class structure repeats the pattern ̅0

_{4}̅1

_{4}̅1

_{4}̅2

_{4}̅3

_{4}̅1

_{4}. Two thirds of the rows in the ring array are even and all are a sum of Fibonacci numbers. Sums of Fibonacci numbers, covering ten, five and three consecutive numbers or number types, had factors of 11, 11 × 31, or 101; (these include specific sets). The Fibonacci number primes all belong to the Class ̅1

_{4}and therefore equal a sum of squares. There is only one unique set of squares with no common factors. The factors found for the sums have a link with Fermat and Mersenne numbers.

**On a new formula for the N-th prime number**

*Original research paper. Page 24*

Krassimir T. Atanassov

Full paper (PDF, 397 Kb) | Abstract

*n*-th prime number as well as for function

*π*(

*n*), determining the number of the prime numbers smaller than

*n*(see, e.g. {1}). In {2} we introduced three new formulae for

*π*(

*n*) and a new formula for the

*n*-th prime number

*p*. Now we shall introduce another — simpler formula for

_{n}*π*(

*n*) and

*p*, following {2}.

_{n}