The Distribution of Addition and Subtraction over Multiplication in Elementary Algebra

A Research Project: The Distribution of Addition over Multiplication in Elementary Algebra. [THIS CONTRIBUTION HAS BEEN REPLACED BY: " The Prime Sequence Problem " In the contribution " The Prime Sequence Problem " an uneven number of primes in the ascending order is added algebraically to its negative summation, thus: Given that the prime sequence is: [3, 5, 7, 11, 13, 17, 19, 23, ... ], noting the absence of the prime, two [2] in the list. The first three primes are written as if they were co-ordinates in a four dimension Euclidean space: [3, 5, 7, - ]. The blank is filled with the integer: - [3 + 5 + 7], which is - 15. The point in 4D is now: [3, 5, 7, - 15]. The next point in 4D space is: [5, 7, 11, - 23]. The next point in 4D is: [7, 11, 13, - 31]. The general point in 4D space is: [p[k - 1], p[k], p[k + 1], - [p[k - 1] + p[k] + p[k + 1]] This sequence of points in 4D space constitutes a polygonal arc in a hyperspace. The purpose of making a zero summation of the co-ordinates is so that the equation: p[k - 1] + p[k] + p[k + 1] - [p[k - 1] + p[k] + p[k + 1] = 0 is always satisfied, noting that an even number of uneven integers has an even summation, remembering that zero is an even integer. Joining successive points and finding the intersections with the planes [axial planes] formed by the axes of co-ordinates enables the dimension to be reduced by unity [because one of the co-ordinates is zero on an axial plane. There is no loss of information in this process. Eventually the dimension can be reduced to 2D and the intercepts in 2D found as a discrete finite function of the gradients. This is an example of a discrete version of the Legendre Transformation. If a regularity could be found in 2D space, then the finite discrete function[s] that represent the prime sequence in 2D could, in principle be extended and the whole process reversed, so allowing the 4D polygonal arc to be augmented. This means that the positions of successor primes located in 4D space could be established. Solving the prime sequence problem then amounts to solving problems in linear algebra and a version of the Legendre transformation, provided that regularity can be found in 2D space.

The reader who is interested in the problem of computing the successive primes is respectfully redirected to the contribution [amongst others]: " The Prime Sequence Problem ".

.SHAWWPG19410425 (talk) 22:01, 20 November 2012 (U.T.C.)SHAWWPG19410425 (talk) 22:35, 1 December 2012 (UTC)