# Volume 10, 2004, Number 1

**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

Modular-ring row structures are developed for squares. In particular, the row structures of even squares within the modular ring Z_{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

An analysis is made of the class and row structures of Fibonacci numbers within the modular ring Z_{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

There are some formulae for the *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*_{n}. Now we shall introduce another — simpler formula for *π*(*n*) and *p*_{n}, following {2}.

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