Check if a number is prime - fast
So far, finding out if a number
n is prime or not was done through basically a linear search of divisors. This means that if we increased n, the number of operations to find out if n is prime or not would increase proportionally (with some constant multiplied). So, if considering the worst-case scenarios (all the cases when the number n was a prime number - 51, 107, etc) - our algorithm was doing approximately n/4 operations to check n for primality.This means that when increasing
n 10 times, the algorithm would perform 10 times more operations. If we increase n 100 times, the algorithm would perform 100 times more operations. This means that our algorithm is performing proportionally to n. Is there a way to do the checking faster? Seems like there is still a lot of redundant checking done.The approach:
Imagine the number
n has a divisor d. This means that
n is divisible by d and both d and d or n itself, so we can check for the smallest of d or This is a huge improvement! Doing
n is a huge difference. Imagine an algorithm that had to work for 10 years (3650 days) to complete a calculation. In case it’s optimized to do n, it would only take it 2 months (61 days) to do so.Are there better algorithms to check the primality of a number?
Yes! You can read more about them on Algorithms for Competitive Programming. There are algorithms that grow (do more operations as
n increases) logarithmically instead of Input
The first line of the input contains a single integer
n (1 ≤ n ≤ ).Output
The program should print
Yes in case n is prime and No otherwise.Examples
Input | Output |
1 | No |
11 | Yes |
353557 | Yes |
34134741 | No |
902468477 | No |
Constraints
Time limit: 2 seconds
Memory limit: 512 MB
Output limit: 1 MB