Volume 10, 2004, Number 1

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.

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.

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}.