partition

1 PLAYING WITH PARTITIONS ON THE COMPUTER (Has appeared on Mathematics and Computer Education) Abdulkadir Hassen Thomas J. Osler Mathematics Department Rowan University Glassboro, NJ 08028 hassen@rowan.edu osler@rowan.edu 1. INTRODUCTION One of the joys of mathematical study is the discovery of unexpected relations. In this paper we explore the strange interplay between partitions and pentagonal numbers. An important function in number theory is pn( ) , the number of unrestricted partitions of the positive integer n, that is, the number of ways of writing n as a sum of positive integers. For example, 4+2+2+1 is a partition of the number 9. The order of the summands is irrelevant here, so 4+2+2+1 is the same partition as 2+2+4+1. In Table 1 we show all the partitions of the numbers from 1 to 5 along with the values ofpn(). Table 1: Partitions of a natural number n n Partitions of n p(n) 1 1 1 2 2, 1+1 2 3 3, 2+1, 1+1+1 3 4 4, 3+1, 2+2, 2+1+1, 1+1+1+1 5 5 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1 7 While it is simple to determine pn() for very small numbers n by actually counting all the partitions, this becomes difficult as the numbers grow. For example, p()10 = 42 , and p()20 = 627 , while p()100 = 190,569,292 . It is the purpose of this 2 paper to show how to write a simple program in BASIC to calculate pn( ) . Along the way we will encounter several nifty mathematical relations. The values of the partition ...