Note: This theorem is widely regarded as having been solved. As of Apr. 2019, this entry is now moved to 'Deprecated'
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. https://en.wikipedia.org/wiki/Four_color_theorem
Computational tests, checking millions of possible configurations, have never found an exception. (note that a formal proof with 5 colors was made in the late 19th century (Heawood 1890) https://en.wikipedia.org/wiki/Percy_John_Heawood
A 'machine generated proof' was described by Georges Gonthier of Microsoft Research Cambridge England, in 2007. see: Asian Symposium on Computer Mathematics, ASCM 2007 Asian Symposium on Computer Mathematics, ASCM 2007: Computer Mathematics pp 333-333
Also see Formal Proof—The Four Color Theorem