An algorithm and a sieve to find prime numbers below 2n for given natural number n

Abstract

Testing the primality of a number is easy for computers when compared to finding the prime factors of a number. Factorization of a number involves division process. This paper aims at simplifying the process of division of a positive odd number. Prime numbers greater than 2 are odd numbers. An odd number x is a prime number if x is not divisible by any prime number less than or equal to square root of x. When the odd number is large then, the process of division becomes tedious for calculators and computers. In this paper, the divisibility of an odd number is known by dividing a smaller number by prime numbers less than or equal to square root of the given odd number. We present certain theorems on the divisibility of sum of two natural numbers by a given natural number. In addition to this, we present an algorithm, and a sieve technique that uses these theorems, to sieve out the composite numbers up to 2n − 1 for a given natural number n. The odd numbers that are not sieved out up to 2n − 1 are prime numbers. Thus, for a given natural number n, we can find prime numbers up to 2n − 1. We have further modified our algorithm to sieve out only odd composite numbers after the multiples of 2 are sieved out.