This paper presents a new method for generating all prime numbers up to a particular number $m\in N,m⩾9,$ by using the set theory. The proposed method is explicit and works oriented in finding the prime numbers in order. Also, we give an efficiently computable explicit formula which exactly determines the number of primes up to a particular number $m\in N,m⩾9.$ For the best of our knowledge, this is the first exact formula given in literature. For the sake of comparison, a unified framework is done not only for obtaining explicit formulas for the well-known sieves of Eratosthenes and Sundaram but also for obtaining exact closed form expression for the number of generated primes using these two sieve methods up to a particular number $m\in N,m⩾9.$ In addition, the execution times are calculated for the three methods and indicate that our proposed method gives a superior performance in generating the primes.

本文提出了一种新的方法来生成所有质数直到一个特定的数 $米\in ñ，米⩾9，$通过使用集合论。所提出的方法是显式的，并且以按顺序查找素数为工作方向。此外，我们给出了一个可有效计算的显式公式，该公式可精确确定素数直至特定数目的数$米\in ñ，米⩾9。$就我们所知，这是文献中给出的第一个确切公式。为了进行比较，我们建立了一个统一的框架，该框架不仅用于为著名的Eratosthenes和Sundaram筛子获得明确的公式，而且还可以使用这两种筛分方法直至特定的数量来获得所生成质数的精确闭合形式表达式。数字$米\in ñ，米⩾9。$ 此外，对这三种方法的执行时间进行了计算，这表明我们提出的方法在生成素数方面具有出色的性能。