We introduce an algorithm that generates primes included in a given interval [ ],

We study values of k for which the interval (kn,(k + 1)n) contains a prime for every n> 1. We prove that the list of such integers k includes 1,2,3,5,9,14 and no others, at least for k ≤ 100,000,000. Moreover, for every known k in this list, we give a good upper bound for the smallest Nk(m), such that if n ≥ Nk(m), then the interval (kn,(k +1)n) contains at least m primes.

We have devised an alternative approach to sifting integers in the sieve of Eratosthenes that helps refine the error term. Instead of eliminating all multiples of a prime number $p<z$ in the traditional sieve method, our approach solely eliminates multiples of $p$ that have the minimum prime factor of $p$. By leveraging the density of integers with the least prime factor $p$ in this sieve technique, we obtain a reduced error term and an upper bound of $\pi(x)$ that accurately reflects the prime number theorem.

We study values of k for which the interval (kn,(k+1)n) contains a prime for every n&gt;1. We prove that the list of such integers k includes k=1,2,3,5,9,14, and no others, at least for k&lt;=50,000,000. For every known k of this list, we give a good upper estimate of the smallest N_k(m), such that, if n&gt;=N_k(m), then the interval (kn,(k+1)n) contains at least m primes.

There are several algorithms to obtain prime numbers. This paper presents an algorithm that can generate all prime numbers without any composite number. Also to consider stopping criterion standard for this algorithm, computer programs can prevent uncontrolled growth of composite numbers set. In fact, the mathematical algorithm stopping criterion standard sieve of Eratosthenes.