The problem
S(r) is the largest n such that {1,…,n} admits an r-colouring avoiding a monochromatic solution of x + y = z. Schur proved S(r) finite in 1916. Known: S(4) = 44 (1918→1996); determine S(5).
The Fifth Schur Number S(5)
historicReferences
History & significance
Schur's theorem launched partition regularity. Each increment demanded exponentially harder search: S(5) candidates hovered at 160 for decades after cumulative partial work (Exoo's lower-bound colourings, Fredricksen & Sweet's S(4)).
The resolution (computer-assisted proof)
Determined 2017 by Marijn Heule: Cube-and-Conquer SAT solving proved every 5-colouring of {1,…,161} yields a monochromatic x+y=z, with verified certificates; combined with Exoo's explicit colouring of {1,…,160}, S(5) = 160. A clean century bookend: Schur's theorem born 1916, its fifth level measured exactly one hundred years later.