Homogeneous Diophantine Equation of Degree Two in NP-Complete

EasyChair Preprint 9354, version 3

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. Certainly, using the Hasse principle we may able to decide whether a homogeneous Diophantine equation of degree two has an integer solution: we are capable to reject an instance when there is no solution reducing the equation modulo p. 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

@booklet{EasyChair:9354,
  author    = {Frank Vega},
  title     = {Homogeneous Diophantine Equation of Degree Two in NP-Complete},
  howpublished = {EasyChair Preprint 9354},
  year      = {EasyChair, 2022}}