A BINARY OPERATION ON THE POSITIVE INTEGERS

We discuss the binary operation * on the positive integers, defined as , in more detail. If you are not familiar with the terms identity, commutatuve, associative and inverse then you should read the relevant sections in these notes before proceding. First of all, there is no left identity element. For if x were such an identity, we would have x*n = n for all n. Now x*n is defined to be the positive integer and so we want to find a solution to for all positive integers n and x fixed. In particular, which is x would have to be 1. Hence x is 1. However, is not 2 and so . There is however a right identity, namely the integer 1. To see this, note that as required. There are no right inverses for any positive integer m larger than 1 since . So the only positive integer that has a right inverse is 1 and every positive integer is a right inverse of 1 (since 1*n = 1). Alternatively, every positive integer has a left inverse, namely 1.

The operation is not commutative since for example 2*3=8 whereas 3*2=9. Similarly, the operation is not associative since 2*(2*3) = 2*8 = 256 and (2*2)*3 = 4*3 = 64. The first few entries of this infinite multiplication table are given below.