0x000b - Sieve of Eratosthenes

Tue, 01 Nov 2011 20:19:54 +0000

The Sieve of Eratosthenes, named after the Greek mathematician Eratosthenes, is one of many known prime number sieves. The algorithm is a fairly good one for finding prime numbers from a list of numbers, say, less than 10 million.

Suppose we need to find all prime numbers below a natural number n > 1. Then, the algorithm is given as:

Create a list of consecutive integers from 2 to n: (2, 3, 4, …, n).
Initially, let p equal 2, the first prime number.
Starting from p, count up in increments of p, as long as the value is less than √n and mark each of these numbers greater than p itself in the list. These numbers will be 2p, 3p, 4p, etc.; note that some of them may have already been marked.
Find the first number greater than p in the list that is not marked; let p now equal this number (which is the next prime).
If there were no more unmarked numbers in the list, stop. Otherwise, repeat from step 3.

When, the above algorithm terminates, all unmarked numbers are prime.

0x0007 - Primality Testing

Sat, 29 Oct 2011 22:58:18 +0000

I’ve been doing some programs on Primality tests, and so far, have implemented some naive algorithms. Though not efficient, these are useful for quick implementations. Here I post some of them:

Method 1:

The simplest primality test is as follows: Given an input number n, check whether any integer m from 2 to n − 1 evenly divides n. If n is divisible by any m then n is composite, otherwise it is prime. The implementation in Python 2.7 in the form of a function is:

Primality on Rohit Jha

0x000b - Sieve of Eratosthenes

0x0007 - Primality Testing