Partitions

Abstract

Until now this book has been concerned primarily with multiplicative number theory, a study of arithmetical functions related to prime factorization of integers. We turn now to another branch of number theory called additive number theory. A basic problem here is that of expressing a given positive integer n as a sum of integers from some given set A, say

$$A = \left\{ {{a_1},{a_2},...} \right\}$$

, where the elements a _i are special numbers such as primes, squares, cubes, triangular numbers, etc. Each representation of n as a sum of elements of A is called a partition of n and we are interested in the arithmetical function A(n) which counts the number of partitions of i into summands taken from A. We illustrate with some famous examples.