Permalink Submitted by Anonymous on December 2, 2012

This was said to be unsolvable by a no less eminent scientist than G.B. F. Riemann. However, if ordered sets of the prime sequence are set out as another sequence:
[3, 5, 7, - 15], [5, 7, 11, - 23], [7, 11, 13, - 31], ... and these sets are treated as points in a four dimensional hyperspace, then the sequence becomes a polygonal arc in 4D. Using linear algebra, finding the direction ratios and intercepts of the line segments produced of the arc, starting from the first two points provides a means of lowering the dimension of the space by unity at a time, but with no loss of information about the prime sequence. This is because on the axis planes, where intercepts are located, one of the co-ordinates is zero and the other remaining co-ordinates become a point in a space of dimension, one less. Finally in 2D space, the intercepts of finite discrete functions of the gradients or higher derivatives might have regularity, or a way of producing regularity might be found. This exercise can be done in any even dimension hyperspace: [4, 6, 8, ... ]
There are four co-ordinates here because of a constraint, that is the sum of an even number of uneven integers is an even integer [zero included]. The solution to the subject appears to reduce to an exercise in linear algebra and a discrete version of the Legendre transformation.

## Computing the next prime up in the sequence of primes.

This was said to be unsolvable by a no less eminent scientist than G.B. F. Riemann. However, if ordered sets of the prime sequence are set out as another sequence:

[3, 5, 7, - 15], [5, 7, 11, - 23], [7, 11, 13, - 31], ... and these sets are treated as points in a four dimensional hyperspace, then the sequence becomes a polygonal arc in 4D. Using linear algebra, finding the direction ratios and intercepts of the line segments produced of the arc, starting from the first two points provides a means of lowering the dimension of the space by unity at a time, but with no loss of information about the prime sequence. This is because on the axis planes, where intercepts are located, one of the co-ordinates is zero and the other remaining co-ordinates become a point in a space of dimension, one less. Finally in 2D space, the intercepts of finite discrete functions of the gradients or higher derivatives might have regularity, or a way of producing regularity might be found. This exercise can be done in any even dimension hyperspace: [4, 6, 8, ... ]

There are four co-ordinates here because of a constraint, that is the sum of an even number of uneven integers is an even integer [zero included]. The solution to the subject appears to reduce to an exercise in linear algebra and a discrete version of the Legendre transformation.