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A prime number is a a positive integer which is divisible only by 1 or itself. Finding whether a number is prime or not is an interesting programming challenge for long time. Moreover there are different menthods and they have different efficiency. In this article we will look at three such methods and judge which one is more efficient in terms of the timing of their execution.

This is a straight forward program where we take every integer from 1 to one less than the given number and keep checking if the number gets divided by any of these. If no number is found which can divide this number then the number is prime.

import time #Function to check Prime Number def check_prime(final_val): if final_val <= 1: return False for divisor in range(2,final_val): if final_val % divisor == 0: return False return True # Track the Start Time StartTime = time.time() #Count the number of prime numbers cnt = 0 for final_val in range(1,10001): x = check_prime(final_val) cnt += x print 'Count of prime numbers till',final_val,'is ', cnt # Track the End Time EndTime = time.time() print 'Time Elapsed is: ', EndTime - StartTime

Running the above code gives us the following result −

Count of prime numbers till 10000 is 1229 Time Elapsed is: 2.3100001812

Mathematically it is also found that it is enough to find the factors till square root of the number for which we are checking. This approach reduces the number of iterations and hence should be faster which we can check as below. The logic to implement this idea is below.

We find out the square root of the number which is being checked for prime value.

We divide the number with each of the value till the square root value starting from value 2, to check if any remainder is left.

If at any step in the above the remainder left is zero, then the number is not prime.

import math import time def is_prime(final_val): # 1 is not a prime number if final_val <= 1: return False i = 2 while i <= math.floor(math.sqrt(final_val)): # Check if any remainders are cerated if final_val % i == 0: return False i += 1 return True # Track the Start Time StartTime = time.time() cnt = 0 for n in xrange(1, 10001): x = is_prime(n) cnt += x print 'Count of prime numbers till',n,'is ', cnt # Track the End Time EndTime = time.time() print 'Time Elapsed is: ', EndTime - StartTime

Running the above code gives us the following result −

Count of prime numbers till 10000 is 1229 Time Elapsed is: 0.0529999732971

In this method we eliminate the non-prime or composite numbers to get the prime numbers till aspecific number. So below are the steps we follow to achieve that.

Make a list of consecutive integers from 2 to that number till which we want find all the prime numbers.

Take the first number i.e., 2 and create a list of all its multiples. Eliminate this list of multiples from the original list but not 2. Repeat this for the next number i.e., 3 and so on for the next numbers. Please note that we are eliminating only the multiples and not the number itself. So 5 or 11 never gets eliminated but 10 and 22 get eliminated.

After all the eliminations the left over list is the list of prime numbers till the asked number.

import time def sieve_method(n): #Create a list of prime numbers prime_number_list = [] for i in range(2, n+1): # Capture the number if it si not in prime list if i not in prime_number_list: print (i) # Add the number to the prime list number if it is a multiple for j in range(i*i, n+1, i): prime_number_list.append(j) # Track the Start Time StartTime = time.time() cnt = 0 print(sieve_method(25)) # Track the End Time EndTime = time.time() print 'Time Elapsed is: ', EndTime - StartTime

Running the above code gives us the following result −

2 3 5 7 11 13 17 19 23 Time Elapsed is: 0.0

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