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 […]
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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 .
The angle sizes of an acute angled triangle can themselves be used to construct another triangle (a bit like turning the triangle inside out). This cannot be done with right or obtuse angled triangles.
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.
The roots of polynomial equations in one variable can be shown easily to satisfy 2 independent linear partial differential equations with respect the equations coefficients. The discriminant also satisfies 2 similar independent LPDE’s.
A method is described for arriving at a power series expression for a real root of a Bring-Jerrard form quintic equation. The method relies on showing that any solution of the quintic must also satisfy a 4th order LDE. The 4th order LDE is easily solvable to give a power series expression for the solution.
An inequality for the area of a triangle A is derived in terms of the perimeter p. This inequality is related to the Weitzenbock inequality and is an improvement on it. The method used also throws up a second triangle inequality. A further link to the Hadwiger-Finsler inequality is also shown. The Ono inequality for acute angled triangles is also proved.
There are a number of inequalities that can be easily derived for the general cuboid. These inequalities connect the Perimeter P, Surface area S and the Volume V. The proof of one of these I have seen elsewhere, but I have not seen all of these published together. These can be easily applied to the general triangle via Heron’s formula. See the ‘Knol’ on triangle inequalities for the applications.
A method is described for solving cubic equations. The roots of a cubic equation are treated as smooth functions of the equations’ coefficients and shown to be solutions to a second order linear differential equation which is easily solvable.
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