Do 05 April 2018 by Samdney
Category: Math
Since I wasn't happy about the final sum in my last post
Primes matrix: Approximation, I think about an alternative way.
We had
$$
x_{\left(a,\dots,b\right),kj} = \lim_{m \rightarrow \infty} \left(\prod_{i=a}^{b} \exp\left(I 2\pi\frac{kx_{i}}{2x_{i} + 1} \epsilon\left(m\right)\right)\right) \delta_{kj} \\
= \lim_{m \rightarrow \infty} \exp\left(\sum_{i = a}^{b} I2\pi\frac{k  x_{i}}{2x_{i} + 1}\epsilon\left(m\right)\right) \delta_{kj}
$$
in which we made the product over all \(\exp\)functions for each \(x_{i}\). Now,
instead we will do the product over the arguments of the \(\exp\)functions
$$
x_{\left(a,\dots,b\right),kj} = = \lim_{m \rightarrow \infty} \exp\left(\prod_{i = a}^{b} I2\pi\frac{k  x_{i}}{2x_{i} + 1}\epsilon\left(m\right)\right) \delta_{kj}
$$
Let's look at the qualities of this product
$$
\prod_{i = a}^{b} \frac{k  x_{i}}{2x_{i} + 1}
$$
and under which conditions we receive integers. From my work
https://github.com/Samdney/primescalc
we already know that we get troubles if at least one of the \(2x_{i} + 1\) is
a divisible number. Hence, we always asume that all our
numbers \(2x_{i} + 1\) are primes.
We receive integer values in the following cases

Case 1: For all \(k\)values which are also solutions for every single
\(\exp\)equation.

Case 2: For all \(k\)values which is a solution for at least one single
\(\exp\)equation and also leads to the trivial solution with
\(x_{\left(1\right),j} = N\left(2x_{\left(2\right),i} + 1\right)\), \(N \in \mathbb{N}\).
For the second case, we take the example of two equations with \(x_{1} = 2\) and
\(x_{2} = 3\)
$$
\frac{k  x_{1}}{2x_{1} + 1} \frac{k  x_{2}}{2x_{2} + 1} = \frac{\left(k  2\right)\left(k  3\right)}{5
\cdot 7}
$$
Here we receive one solution for \(k = 37\), \(\frac{35 \cdot 34}{35} = 34\), which is also a solution for
\(\frac{37  2}{5} = 7\) and an other solution for \(k = 38\), \(\frac{36 \cdot
35}{35} = 36\) which is also a solution for \(\frac{38  3}{7} = 5\). We see that
\(k\) leads to the trivial case in which \(x_{j}\) of one single \(\exp\)equation is equal to the prime value of an other single \(\exp\)equation or the product of primes of several single \(\exp\)equations.