Volume 5, 1999, Number 3

A modular ring ℤ₈ is described as an expanded form of ℤ₄ in which the parities of the rows within the four classes of ℤ₄ are distinguished. This octic or ‘chess’ ring is used to analyse the structure of rows characterising powers xⁿ within the eight classes of ℤ₈. With n even, three classes contain the squares, x², but only two contain xⁿ when n > 2. With n odd, even xⁿ are confined to 0₈, but odd numbers, xⁿ, have the same class as x. The rows containing squares are related to first order recurrences, while rows containing xⁿ, n > 2, are related to third and higher order recurrences. Because of this it is conjectured that the sum or difference of xⁿ, yⁿ cannot fit into a row which is characteristic of a power and hence cannot equal a power when n > 2.

By using the geometry of a pythagorean triangle with a circle inscribed, it can be proved by a simple geometric proof that the area of such a triangle can never be a square. The class structure of the modular ring Z₄ can be used to illustrate the result for various Pythagorean triples.