Abstract
For a semiprime that consists in two distinct odd prime divisors, this article makes an investigation on the distribution of the small divisor by analyzing the divisor-ratio that is calculated by the big divisor divided by the small one. It proves that, the small divisor must be a divisor of an odd integer lying in an interval that is uniquely determined by the divisor-ratio, and the length of the interval decreases exponentially with the increment of the ratio. Accordingly, a big divisor-ratio means the small divisor can be found in a small interval whereas a small divisor-ratio means it has to find the small divisor in a large interval. The proved theorems and corollaries can provide certain theoretical supports for finding out the small divisor of the semiprime.
Highlights
A semiprime is an odd composite number N that has exactly two distinct prime divisors, say p and q, such that 3 ≤ p < q
For a semiprime that consists in two distinct odd prime divisors, this article makes an investigation on the distribution of the small divisor by analyzing the divisor-ratio that is calculated by the big divisor divided by the small one
The small divisor must be a divisor of an odd integer lying in an interval that is uniquely determined by the divisor-ratio, and the length of the interval decreases exponentially with the increment of the ratio
Summary
A semiprime is an odd composite number N that has exactly two distinct prime divisors, say p and q, such that 3 ≤ p < q. Finding an effective approach to factorize a semiprime still remains a research work for researchers all over the world, as stated in (Duta, 2016) and (WANG X,2017 RSA). In February 2017, WANG X (WANG X, 2017 Genetic) introduced an approach that can exactly locate the divisors of a composite odd number in respectively definite intervals and proposed an algorithm that can factorize composite odd integers. Since the algorithm is still slow in factoring big semiprimes, as stated in (WANG X, 2017 RSA), this article, in order to know clearly the semiprimes and to develop more efficient algorithms to factorize a big semiprime, follows the ideas in (WANG X, 2017 Genetic) and (WANG X, 2017 RSA) to make an investigation on distribution of the semiprime’s divisor.
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