the dictionary definition of prime numbers gives us a clear idea about prime number finding algorithm; ‘–noun Mathematics.
a positive integer that is not divisible without remainder by any integer except itself and 1, with 1 often excluded: The integers 2, 3, 5, and 7 are prime numbers. ‘ according to http://dictionary.reference.com/browse/prime+number.
hence the most primitive algorithm is dividing the number by all the positive numbers between 2 and the number-1 and waiting for non-zero reminders from all.
however it is not really the most practical way.
first of all it is obvious that any numbers except for 2 cannot be divided without reminder by its decrement. this way of thinking leads us to question the range of the numbers to be used as divider. logically the limit should be between 2 and square root of the number. the explanation is simple; lets assume that the number is not prime and we divide it with a number bigger than its square root, in that case the result will be a number between 2 and the number’s square root, which we should have already tested (since we start the division test in order starting from two).
secondly the test should be done only with prime numbers, not all. the reason is quite simple, if X divides Y then all the multipliers of X also divides Y. think about dividing 600 by 30; 600 can also be divided by 2, 3 and 5 which are the multipliers of 30.
lastly there is no point of testing even numbers as they already can be divided by 2. by doing that, for finding prime number in a range, we also prevent the division by 2 tests.
here is the python code for finding prime numbers in the range of 2 to 1’000’000, takes aroun 5.4 seconds on my laptop.
from math import sqrt import time t1 = time.time() primenumbers=[2,3,5] for i in range(7,1000000,2): sq = sqrt(i)+1 for j in primenumbers: if sq < j: primenumbers.append(i) #print i, len(primenumbers) break if i % j == 0: break print len(primenumbers) t2 = time.time() print (t2 - t1)