Talk:Formula for primes

I have LaTeX translated Venugopalan's formula. It is given below:

${\displaystyle P_{n+1}^{2}{\text{ = 1 + }}\left|{\text{ }}-{{\log }_{X}}\left({\scriptscriptstyle 1\!/\!{}_{2}}{\text{ }}\sum \limits _{{{a}_{1}}=1}^{{{p}_{1}}-1}{...\sum \limits _{{{a}_{n}}-1}^{{{p}_{n}}-1}{\sum \limits _{b=1}^{n}{{X}^{-{{\left(\sum \nolimits _{1}^{n}{{{a}_{i}}{{Q}_{i}}-bQ}\right)}^{2}}}}}}-{\text{ 1/X }}\right)\right|}$

He says X could be any number (not an integer). I guess, we could use Euler's constant "e" in place of X so that ${\displaystyle Lo{{g}_{x}}}$ will be Natural logarithm for ease of computation. — Preceding unsigned comment added by 64.121.225.161 (talk) 15:20, 13 August 2014

For P differs from zero, P is the first Twin prime between ${\displaystyle {{p}_{n}}}$ and ${\displaystyle p{{_{n}^{2}}_{}}}$ and it is given by the formula below:

${\displaystyle P{\text{ = Y}}{\sqrt {\left(1-\left|{{\sin }^{2}}\left({\frac {\pi }{2}}-\left[{{2}^{-\left({{Y}^{2}}\_{{p}^{4}}_{n}\right)}}\right]\right)\right|\right)}}}$

where,

${\displaystyle {{Y}^{2}}={\text{ }}\left(1+\left|-{{\log }_{x}}\left({\frac {1}{2}}\sum \limits _{{{a}^{'}}_{1}\neq {\text{ }}{{a}_{1}}=1}^{\left({{p}_{1}}-1\right)}{...}{\text{ }}\sum \limits _{{{a}_{n}}^{'}\neq {{a}_{n}}=1}^{\left({{p}_{n}}-1\right)}{\sum \limits _{b=1}^{n}{{{X}^{-{{\left(\sum \nolimits _{1}^{n}{{{a}_{i}}{{Q}_{i}}-bQ}\right)}^{2}}}}\_\left({\frac {1}{X}}\right)}}\right)\right|\right)}$

${\displaystyle X=any{\text{ }}positive{\text{ number, }}\left({{a}^{'}}_{1},{{a}^{'}}_{2},...,{{a}^{'}}_{n}\right)be{\text{ unique solution of}}}$

${\displaystyle -{\text{2}}\equiv {\text{ }}\sum \nolimits _{1}^{n}{{a}_{i}}{{Q}_{i}}{\bmod {Q}}{\text{ with 0}}\leq {{\text{a}}_{i}}\leq \left({{p}_{i}}-1\right){\text{ ,}}}$

${\displaystyle i=1,2,3..,n{\text{ and n}}>1}$

${\displaystyle {\begin{aligned}&{\text{For any integer x in }}\left({{p}_{n}},p_{n}^{2}\right),{\text{ the number of primes not greater than x,}}\\&{\text{is given by the formula below:}}\\&\\&\prod {\left(x\right)}={\text{ }}{\frac {1}{2\pi }}\int \limits _{0}^{2\pi }{\prod \limits _{i=1}^{n}{\frac {\sin \left(\left(\left(Q-{{Q}_{i}}\right)/2\right)\theta \right).\sin \left(\left(n-1\right)Q\theta /2\right).\sin \left(x\theta /2\right)\cos \left(\left(x+1\right)\theta /2\right)}{\sin \left({{Q}_{i}}\theta /2\right).\sin \left(Q\theta /2\right)\sin \left(\theta /2\right)}}}d\theta +n\\&where,\\&{{Q}_{i}}{\text{ = }}Q/{{p}_{i}}{\text{ for }}i=1,2,3,...,n\\\end{aligned}}}$

${\displaystyle {\begin{aligned}&{\text{If }}\prod \nolimits _{2}{\left(x\right){\text{ represents }}}{\text{the number of twin-primes not exceeding integer x }}\\&{\text{for }}{{\text{p}}_{n}}\langle {\text{ x }}\langle p_{n}^{2}{\text{ , then,}}\\&\prod \nolimits _{2}{\left(x\right)}{\text{ = }}{\frac {1}{2\pi }}\int \limits _{0}^{2\pi }{\sum \limits _{{{a}_{1}}\neq a_{1}^{'}}^{}{...\sum \limits _{{{a}_{n}}\neq a_{n}^{'}}{\sum \limits _{b=1}^{\left(n-1\right)}{\sum \limits _{t=1}^{x}{\cos \left(\sum \nolimits _{1}^{n}{{{a}_{i}}{{Q}_{i}}-bQ}\right)}}}}}\theta \cos \left(t\theta \right){\text{ }}d\theta +\prod \nolimits _{2}{{p}_{n}}\\&where,\\&Q=\prod \limits _{i=1}^{n}{{p}_{i}}{\text{ for }}i=1,2,3...,n{\text{ and }}{{Q}_{i}}=Q/{{p}_{i}}\\&{\text{Since all twin-primes upto and including }}{{\text{p}}_{n}}{\text{ are known }}\prod \nolimits _{2}{\left({{p}_{n}}\right)}{\text{ is known}}{\text{.}}\\&{\text{Twin-prime conjecture follows immediately if}}\prod \nolimits _{2}{\left(x\right)}{\text{ is greater than }}\\&\prod \nolimits _{2}{{p}_{n}}{\text{ for any integer x}}{\text{.}}\\\end{aligned}}}$