Prime Numbers

Problem 1 - Determine if the give number is prime. What is the fastest way to do this.
Problem 2 - Find all the prime numbers between given numbers X & Y. What is best way
                     to do this?
Problem 3 - Find all the prime numbers less than or equal to given number N. What is the
                     best way to do this?

So can we see the best code to solve the above problems.
All 3 problems can have separate code & approach but it should be the best implementation to solve the particular problem.

Please share the best implementation here to help the community.