SAT is as Hard as Solving Homogeneous Diophantine Equation of Degree Two

EasyChair Preprint no. 9354, version 9

5 pagesDate: March 15, 2023

Abstract

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. In mathematics, a Diophantine equation is a polynomial equation, usually involving two or more unknowns, such that the only solutions of interest are the integer ones. A homogeneous Diophantine equation is a Diophantine equation that is defined by a homogeneous polynomial. Solving a homogeneous Diophantine equation is generally a very difficult problem. However, homogeneous Diophantine equations of degree two are considered easier to solve. We prove that this decision problem is actually in NP-complete under the constraints that all solutions contain only positive integers which are actually residues of modulo a single positive integer.

Keyphrases: Boolean formula, completeness, complexity classes, polynomial time