Question

The following algorithm checks whether a number is prime:

Given a number n,loop over all numbers smaller than n and check whether they divide n. 
If one of them divides n, answer no. Otherwise, answer yes.

Now, I have to analyse the number of division operations performed by the algorithm as a function of the length of its input in the following two cases:

1) The number is encoded in unary (i.e, 4 is 1111). How do I show that the number of divisions is polynomial?

2) The number is encoded in binary (i.e, 4 is 100). How do I show that the number of divisions is exponential?

Answer 1

Suppose we have n 1‘s strung together (notated 1^n). n is the length of our input, obviously. We will divide all the integers from 11, 111, … ,1^(n-1) into 1^n. How many numbers will you be dividing into 1^n, as a function of n? Is this a polynomial?

Note that it takes log_2(x) (log base 2 of x) bits to represent x, approximately, in binary. Also note that we will be performing x-2 divisions (2, 3, 4, 5, … , x-1 will be divided into x). So, for log_2(x) bits we use x-2 divisions. Suppose, instead, that we let n be the size of our input. So we have n = log_2(x). How many divisions will we take, then, in terms of a function of n?