Parity and Primality of Catalan Numbers

Abstract

Catalan numbers, like Fibonacci and Lucas numbers, appear in a variety of situ ations, including the enumeration of triangulations of convex polygons, well-formed sequences of parentheses, binary trees, and the ballot problem [l]-[5]. Like the other families, Catalan numbers are a great source of pleasure, and are excellent candidates for exploration, experimentation, and conjecturing. They are named after the Belgian mathematician Eugene Catalan (1814-1894), who discovered them in his study of well-formed sequences of parentheses. However, Leon hard Euler (1707-1783) had found them fifty years earlier while counting the number of triangulations of convex polygons [3]. But the credit for the earliest known discov ery goes to the Chinese mathematician Antu Ming (ca. 1692-1763), who was aware of them as early as 1730 [6]. In 1759 the German mathematician and physicist Johann Andreas von Segner (1707-1777), a contemporary of Euler, found that the number Cn of triangulations of a convex polygon satisfies the recursive formula