Consider a ‘square root’ function defined by f:k -> √(1+6k), k∈R, k≥0. It is easily shown that f(2n(3n+1)) = 6n+1, n≥0, and f(2n(3n-1)) = 6n-1, n≥1/3. Clearly then f(2n(3n+1)) = 6n+1 and f(2n(3n-1)) = 6n-1, n∈ω. The set of primes ≥ 5 ⊂ {6n+1: n∈ω} ∪ {6n-1: n∈ω} and so f() has an infinite number of ‘integer’ points including all of those […]

This article explores a link between a prime number ≥ 5 and a Pythagorean triple containing such a prime.

All Primitive Pythagorean Triples can be shown to belong to a vine-like structure, with a ‘root’, a horizontal ‘main stem’, and binary branching sequences which are seeded from the main stem triples and every other triple below it.

Primitive Pythagorean Triples, or PPT’s, are non-trivial integer solutions to the Pythagorean equation z2 = x2 + y2, where we assume that 0<x<y<z and that x, y and z are co-prime. Euclid’s formulae allow PPT’s to be generated using 2 (positive integer) parameters, usually denoted by m and n. The PPT’s (usually) take the form […]

There are many proofs of this inequality; this is only one of these. There is also an application of the AGM to the sequence n^1/n.

Some initial thoughts on the ‘Fermat’ equation lead on to a detailed analysis of Primitive Pythagorean Triples (PPT’s). A vine-like structure is exposed having a single main stem sequence and binary branching structures of PPT sequences. Every PPT belongs to either the main stem sequence or at least one of the binary branching sequences. A simple method of generating sets of PPT’s is also outlined.

A simple method is described for constructing periodic functions which seem to have a behaviour (shape) similar to that of the periodic solutions to the Van der Pol equation .

Triangle space is the set of all triples (a,b,c) representing the lengths of the sides of a triangle. This space has some interesting properties which can be explored using straightforward mathematical language.

Squares (of whole numbers) have some interesting properties, e.g. most of these can be represented as differences of squares. Other powers (of whole numbers), e.g. cubes, can also be represented as differences of squares of whole numbers.