Other problems discussed include the Goldbach Conjecture, the

Four Color Theorem, the Kepler Conjecture, the Mordell Conjecture, the Riemann Hypothesis, the Poincare Conjecture, the P/NP Problem, the Navier-Stokes Equation, the Mass Gap Hypothesis, the Birch-Swinnerton-Dyer Conjecture, and the Hodge Conjecture.

The four color theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.

In 1976, Kenneth Appel and Wolfgang Haken published their proof of the four color theorem. It was the first major theorem to be proved using a computer.

In this paper, with the help of Neutrosophy and Quad-stage Method, the proof for negation of "the four color theorem" is given.

The process of negating "the four color theorem" with Neutrosophy and Quad-stage method, and deriving "the two color theorem" and "the five color theorem" to replace "the four color theorem", can be divided into four stages.

About these aspects, especially the brilliant accomplishments of proving "the four color theorem" with computer, many discussions could be found in related literatures, therefore we will not repeat them here, while the only topic we should discuss is finding the shortcomings in the existing proofs of "the four color theorem".

For the different proofs of "the four color theorem", we can name the results as "the AppelHaken's four color theorem", "the Chen Jianguo's four color theorem", and the like.