Novel Aspects Of The Global Regularity Of Primes

For any $m=3(2n+1)withn\in {\mathbb{N}}^{\ast },$ the prime counting function $\pi (m)=4+\mid A_4(m)\mid +2\mid A_6(m)\mid$ where $A_6(m)$ and $A_4(m)$ are the sets of Twin Primes and “Isolated” Primes, below $m$, respectively. $T(m)=1+\mid A_2(m)\mid +\mid A_4(m)\mid +\mid A_6(m)\mid$ is the number of consecutive odd composite numbers (COCONs) below $m\mathrm{.}$ $A_2(m)$ is the set of COCONs, below $m$ and distant by 2. With $m$ odd, $\pi (m)$, $T(m)$ and $\mid A_2(m)\mid =\frac{m-9}{2}-3\mid A_6(m)\mid -2\mid A_4(m)\mid$ lead to $4\mid A_6(m)\mid +7=m-2(T(m)+\mid A_4(m)\mid )\mathrm{.}Hence0<1-\frac{2}{m}(T(m)+\mid A_4(m)\mid )<1.$ Thus the non-empty unique set $S\equiv \{1-\frac{2}{l}(T(l)+\mid A_4(l)\mid )suchthatl=3(2k+1)withk\in {\mathbb{N}}^{\ast }\}\subset \mathbb{R}$ is bounded. Therefore $Inf\left(S\right)$ exists, is unique and finite. By definition, $Inf(S)>0.$ $\mathbb{Q}$ dense in $\mathbb{R}$ also guarantees the latter. S and $Inf(S)$ are independent of $m\mathrm{.}$ We then introduce $\alpha =Inf\left(S\right):$ $\alpha$ is independent of $m\mathrm{.}$ We then have $0<\alpha \le 1-\frac{2}{m}(T(m)+\mid A_4(m)\mid )$ for any m as above. Therefore, $4\mid A_6(m)\mid +7\ge \alpha m,$ with $\alpha >0$ and independent of $m\mathrm{.}$ Hence ${\displaystyle \underset{m⟶+\mathrm{\infty }}{\lim }\mid 2A_6(m)\mid =+\mathrm{\infty }\mathrm{.}}$ Similarly, ${\displaystyle \underset{m⟶+\mathrm{\infty }}{\lim }\mid A_4(m)\mid =+\mathrm{\infty }\mathrm{.}}$