Permalink Submitted by Anonymous on December 2, 2012

Further to my comment that the sequence of primes can be thought of as a series of points that are solutions to an equation such as [p[1] + p[2] + p[3] - [p[1] + p[2] + p[3]] = 0, where the co-ordinates are: [p[1], p[2], p[3], - [p[1] + p[2] + p[3]] forming a polygonal arc in a hyperspace of even dimension: [2, 4, 6, 8, ... ], four in this case, the problem being treated as an exercise in linear algebra [the dimension 2D case is degenerate] and a discrete version of the Legendre Transformation, since there is no loss of information in lowering the dimension, it should be possible to reconstruct the 4D, 6D, ... arc from discrete 2D finite functions, derived from the hyperspace in the first instance. If the 2D functions of intercept vs gradient or higher derivatives are regular, then these 2D functions can be augmented and then reconstructed in the hyperspace so allowing more points in the hyperspace to be found and hence more prime numbers computed. In the case of a 6D space, the sequence would read: [3, 5, 7, 11, 13, - 39], [5, 7, 11, 13, 17, - 53] , ... Additionally, the fact that uneven integers differ by a multiple of two and the two categories: [4k + 1] and [4.k + 3] can be built into the maths. I do not know that this approach has ever been used elsewhere.

## The Prime Sequence Problem

Further to my comment that the sequence of primes can be thought of as a series of points that are solutions to an equation such as [p[1] + p[2] + p[3] - [p[1] + p[2] + p[3]] = 0, where the co-ordinates are: [p[1], p[2], p[3], - [p[1] + p[2] + p[3]] forming a polygonal arc in a hyperspace of even dimension: [2, 4, 6, 8, ... ], four in this case, the problem being treated as an exercise in linear algebra [the dimension 2D case is degenerate] and a discrete version of the Legendre Transformation, since there is no loss of information in lowering the dimension, it should be possible to reconstruct the 4D, 6D, ... arc from discrete 2D finite functions, derived from the hyperspace in the first instance. If the 2D functions of intercept vs gradient or higher derivatives are regular, then these 2D functions can be augmented and then reconstructed in the hyperspace so allowing more points in the hyperspace to be found and hence more prime numbers computed. In the case of a 6D space, the sequence would read: [3, 5, 7, 11, 13, - 39], [5, 7, 11, 13, 17, - 53] , ... Additionally, the fact that uneven integers differ by a multiple of two and the two categories: [4k + 1] and [4.k + 3] can be built into the maths. I do not know that this approach has ever been used elsewhere.