# Polygonal Numbers > EYPHKA! num = (triangle) + (triangle) + (triangle) > > Carl Friedrich Gauss ## Introduction ### Triangular numbers T0 = 0 T1 = 1 . T2 = 3 . . . T3 = 6 . . . . . . (...) ### Square numbers S0 = 0 S1 = 1 . S2 = 4 .. .. S3 = 9 ... ... ... (...) ### General case The formula for the n-th s-gonal number is P(s,n) = [(s-2)*n^2 - (s-4)*n] / 2 ## Fermat polygonal number theorem ------------------------------------------------------------------ Thm. Every positive integer is a sum of at most n n-gonal numbers. ------------------------------------------------------------------ For example, 27 = 21 + 6 + 0 (Triangular) = 16 + 9 + 1 + 1 (Square) = 22 + 5 + 0 + 0 + 0 (Pentagonal) = 15 + 6 + 6 + 0 + 0 + 0 (Hexagonal) ... n=4 case is especially famous. It is called Lagrange's four-square theorem.