2 research outputs found

    A Novel Algorithmic Approach using Little Theorem of Fermat For Generating Primes and Poulet Numbers in Order

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    Computer encryption are based mostly on primes, which are also vital for communications. The aim of this paper is to present a new explicit strategy for creating all primes and Poulet numbers in order up to a certain number by using the Fermats little theorem. For this purpose, we construct a set C of odd composite numbers and transform Fermats little theorem from primality test of a number to a generating set Q of odd primes and Poulet numbers. The set Q is sieved to separate the odd primes and the Poulet numbers. By this method, we can obtain all primes and Poulet numbers in order up to a certain number. Also, we obtain a closed form expression which precisely gives the number of primes up to a specific number. The pseudo-code of the proposed method is presented

    A new explicit algorithmic method for generating the prime numbers in order

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    This paper presents a new method for generating all prime numbers up to a particular number m∈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∈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∈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
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