P versus NP

$P$ versus $NP$ is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is $P$ equal to $NP$? It was essentially mentioned in 1955 from a letter written by John Nash to the United States National Security Agency. However, a precise statement of the $P$ versus $NP$ problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. To attack the $P$ versus $NP$ question the concept of $\mathit{\text{coNP-completeness}}$ has been very useful. A well-known $\mathit{\text{coNP-complete}}$ problem is $3UNSAT$. In $3UNSAT$, it is asked whether a given Boolean formula $\varphi$ in $3CNF$ is unsatisfiable.

In this paper, we consider the problem of computing the sum of the weighted densities of states of a Boolean formula in $3CNF$. Given a Boolean formula $\varphi$, the density of states $n(E)$ counts the number of truth assignments that leave exactly $E$ clauses unsatisfied in $\varphi$. The weighted density of states $m(E)$ is equal to $E\times n(E)$. The sum of the weighted densities of states of a Boolean formula in $3CNF$ with $m$ clauses is equal to $\sum _{E=0}^{m}m(E)$. We prove that we can calculate the sum of the weighted densities of states in polynomial time. 

Diophantine equations of the form $\sum _{E=0}^{m}E\times x_E=c$ are solvable in polynomial time for arbitrary values of $m$. We can apply this Diophantine equation such that $c$ is the value of the sum of the weighted densities of states from a Boolean formula $\varphi$ in $3CNF$, $E\times x_E$ is the value of each term $E\times n(E)$, where $n(E)$ corresponds to the unknown value of $x_E$. Hence, we only need to check that $\sum _{E=1}^{m}E\times x_E=c$ has its solutions when $\mathrm{\forall }1\le E\le m:x_E>0$ and $x_E>x_{E+1}$ such that $\sum _{E=1}^m{x}_E\ge 2^n$ is always true, where $n$ is the number of variables in $\varphi$. Certainly, if the Boolean formula $\varphi$ has $n$ variables, then $\varphi$ has exactly $2^n$ possible truth assignments. In this way, we are able to check whether the Boolean formula $\varphi$ is unsatisfiable and obtain a solution for the problem $3UNSAT$ in polynomial time. If any $\mathit{\text{coNP-complete}}$ problem is in $P$, then every $NP$ problem can be decided by a polynomial time algorithm. Consequently, we show the complexity class $P$ is equal to $NP$.