# Volume 10, 2004, Number 1

Volume 10 ▶ Number 1 ▷ Number 2Number 3Number 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 Z6 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 Z4. It is found that the class structure repeats the pattern ̅04 ̅14 ̅14 ̅24 ̅34 ̅14. 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 ̅14 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 pn. Now we shall introduce another — simpler formula for π(n) and pn, following {2}.

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