# Prime Number Distribution Series

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For a long time, when all the prime numbers up-to some given number <math>x</math> were evaluated, it was expected that its distribution or the count off can be represented by a simple analytical function. The distribution of prime numbers is indeed be a pattern related phenomenon but the means that pattern has been sought is historically misguided or ill-advised, according to Yoldas Askan, a British scientist and mathematician. In his paper, Askan challenges some of the fundamental understanding of Prime Numbers and reconsiders these definitions, and ultimately arrives at his analytical formula. In his view, there is no great deal about functions that are approximations because there can be infinitely many of these derived but these will be suitable or good approximations only at a certain number interval. Askan claims that the beautiful thing about the Prime Number distribution is that there will be no analytical function [of any complexity] that will compute and provide exact values for prime counting function, <math>\pi(x)</math>, other than the Prime Number Distribution Series <math>(PNDS)</math> which is provided as follows,

- <math>\pi(x) = (\pi'(x)_n \times 3) - \frac {3}{2} + \frac {-1^{\pi'(x)_n}}{2}</math>

where,

- <math>\pi'(x)_n = \left (\frac {\left ((S_x \times S_r) + \frac {3}{2} - \frac {-1^{(s_x \times s_y + D)}}{2} \right )}{3} \right ) + (i\times (2\times S_x)) + (j\times ((2\times S_y) + (i\times ((2 \times (S_x + E))))))</math>

<math>n = 0..4</math>

<math>S_x</math> and <math>S_y</math> are base primes, <math>(5 \times 5)</math>, <math>(5 \times 7)</math>, <math>(7 \times 7)</math> and <math>(7 \times 11)</math>.

<math>i, j = 0, 1, 2, \ldots</math>

<math>D</math> is a constant equal to either <math>0</math> or <math>1</math> such that provides for integer solutions to <math>\pi(x)'</math>

<math>E</math> is a constant equal to <math>\pm 1</math> (positive for <math>\pi'(x)_1</math> and <math>\pi'(x)_2</math> and negative for <math>\pi'(x)_3</math> and <math>\pi'(x)_4</math>).

To prove that the formula is accurate and not a sieving tool, unlike all current precise prime counting techniques, Askan provides the software code that implements the <math>(PNDS)</math>^{[1]}^{[2]}. Furthermore, by exploiting unpublished and novel techniques in combinatory mathematics, Askan can count <math>\pi(e^{25})</math> in less than <math>10</math> seconds on an average personal computer.

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