About the actual number of primes (3)

PRIME NUMBER SERIES 14

THE PRIME NUMBER FUNCTION $\pi(n_{min},n_{max})$

The prime function $\pi(n_{min},n_{max})$ is defined as the number of all primes p that lie between $2n_{min}+3$ and $2n_{max}+3$ .

The function $\pi(n_{min},n_{max})$ returns from $n_{min}\ge 1$ to $n_{max}\ge n_{min}$ ∈ ℕ the exact number of all primes that exist in the chosen range. Thus, it is not an approximation, but the actual number.

Using

$\pi(n_{min})=2+\sum\limits_{n=1}^{n_{min}}t(n)$

$\pi(n_{max})=2+\sum\limits_{n=1}^{n_{max}}t(n)$

according to paper 8, THE EXACT NUMBER OF ALL PRIM NUMBERS UP TO A SELECTED VALUE, $\pi(n_{min})$ and $\pi(n_{max})$ can be calculated. Taking the difference, we get

$\pi(n_{min},n_{max})=\sum\limits_{n_{min}}^{n_{max}}t(n)$

Legend:

$\sum\limits_{n_{min}}^{n_{max}}t(n)$ = sum of all outcomes of t(n) from $n_{min}$ to $n_{max}$ inclusive.

Proof:

Using the deterministic test function t(n), find t(n) for each n from $n_{min}$ to $n_{max}$ and then add them together. Since t(n) returns only the values 0 for non-prime and 1 for prime, only the number of primes from $n_{min}$ to $n_{max}$ are summed. Since 0 may not be set in t(n) for n, $n_{min}\ge 1$ must be.

What needed to be proved.

Munich, March 10, 2023

Gottfried Färberböck

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THE PRIME NUMBER SERIES

About the actual number of primes (3)