Primes matrix: Approximation 2

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Since I wasn’t happy about the final sum in my last post Primes matrix: Approximation, I think about an alternative way.

We had

$$ x_{\left(a,\dots,b\right),kj} = \lim_{m \rightarrow \infty} \left(\prod_{i=a}^{b} \exp\left(I 2\pi\frac{k-x_{i}}{2x_{i} + 1} \epsilon\left(m\right)\right)\right) \delta_{kj} \\ = \lim_{m \rightarrow \infty} \exp\left(\sum_{i = a}^{b} I2\pi\frac{k - x_{i}}{2x_{i} + 1}\epsilon\left(m\right)\right) \delta_{kj} $$

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.