%0 DATA
%A Frank, Vega
%D 2018
%T Quadratic Congruences on Average Case
%U https://figshare.com/articles/Quadratic_Congruences_on_Average_Case/6118847
%R 10.6084/m9.figshare.6118847.v1
%2 https://ndownloader.figshare.com/files/11027633
%K Complexity Classes
%K Completeness
%K Theory of Numbers
%K Polynomial Time
%K certificates
%X 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? This question was first mentioned in a letter written by John Nash to the National Security Agency in 1955. However, a precise statement of the P versus NP problem was introduced independently in 1971 by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity class is NP-complete. To attack the P versus NP question the concept of NP-completeness has been very useful. The Quadratic Congruences is a known NP-complete problem. We show this problem can be solved in polynomial time for the average case. It is true that Hamilton cycle and some NP-complete problems could be solved in average case over inputs. However, this algorithm, in the same way as Quicksort, is polynomial for a large amount of inputs because of the infinite set of elements that cannot be solved in polynomial time is infinitesimal.