Influence of Divisor-ratio to Distribution of Semiprime ’ s Divisor

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.


Introduction
A semiprime is an odd composite number N that has exactly two distinct prime divisors, say p and q, such that 3 ≤ p < q.Factorization of the semiprimes has been an active topic in the world since the RSA challenge was published due to the close relationship between the semiprimes and the RSA numbers, as were introduced books (Surhone, 2011 and2013).It is believed that, a valid and successful method to factorize a large semiprime rapidly might be a key to open the door of solving the difficult problem of integer factorization.It is known that the GNFS is regarded to be the fastest known general-purpose method to factorize large integers, but the vast amounts of memory that it requires in the computation leaves less chance for an ordinary computer to perform it (Wang Q, 2016).Accordingly, 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 fact, there have been many papers to investigate new effective approaches.For example, Silva J (Silva J, 2010) gave an approach to factorize the semiprime of two equal-sized divisors, Wilson K E (Wilson K E, 2011) attempted factoring semiprimes using PG 2 N prime graph multiagent search, Kloster K ( Kloster K, 2011) and Kurzweg U H (Kurzweg U H, 2012) tried to factorize a semiprime n by estimating the Euler's totient ϕ(n) , Verkhovsky B S (Verkhovsky B S, 2012) attempted to do the factorization of semiprimes based on analysis of a sequence of modular elliptic equations, Grosshans F and his partners ( Grosshans F, 2015) imagined to factorize the safe semiprimes with Quantum computer, Khadir (Khadir, 2016) experimented factoring multi-power RSA moduli with primes sharing least or most significant bits, Zhang H (Zhang H, 2013) and Zheng M (Zheng M, 2017) tested respectively factorization with small prime difference.It can see from these literatures that, a better approach is still in need because there has no report to factorize a big semiprime on an ordinary computer except Kurzweg (Kurzweg U H,2018) who factorized a semiprime of 40 decimal digits with the help of MAPLE in April,2018.
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.Nevertheless,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.By analyzing the divisor-ratio k = q/p of a semiprime N = pq with 3 ≤ p < q, this paper proves several theorems and corollaries to discover p's distribution, as presented in the following sections.Section 2 lists the preliminaries for the later sections; section 3 proves the mathematical foundations.

Preliminaries
This section lists the preliminaries that include definitions, symbols and lemmas, which are necessary for later sections.

Symbols and Notations
In this whole article, a semiprime N = pq means p and q are both odd prime numbers and 3 ≤ p < q.An odd interval [a, b] is a set of consecutive odd numbers that take a as the lower bound and b as the upper bound; for example, [3, 11] = {3, 5, 7, 9, 11}.Symbol ⌊x⌋ is the floor function, an integer function of real number x that satisfies ⌊x⌋ ≤ x < ⌊x⌋+1; symbol GCD means the greatest common divisor; symbol T N means a valuated binary tree that is rooted by N and symbol N (k, j) is a node at the j-th position on level k of T N , as defined in [?]; when m > 0, e b m = N (m+1,2 m−1 −1) = 2 m N − 1 is the rightmost node on level m in the left branch of T N , e p m = 2 m N − p and e q m = 2 m N − q are respectively the first p's multiple-node and the first q's multiple-node that are left to e b m ; by default, , where p, q and α are positive integers with 3 ≤ p < q and α > 2; let symbol e b m+i be the rightmost node on level m + i in the left branch of T N , symbols e p m+i and e q m+i be respectively the first p's-multiple-node and the first q's multiple-  Lemma 4 (See in (WANG X, 2018 Integer Function)) Let a be a given positive constant real number and x be a variable on (0, ∞).Define d(x) and its first order difference ∆ x by (2) For a given positive real number ω, it holds Furthermore, for given real numbers αand β with 0 ≤ α < 1 and β > 1, there always exists an x 0 such that, when x > x 0 it holds Lemma 5 (See in (WANG X, 2018 Integer Function)) Suppose k 1 and k 2 are positive integers with k 1 < k 2 , a is real with a > max(k 1 + 1, k 2 + 1) and b is a constant real number; let s 1 =

Main Results and Proofs
Theorem 1 Suppose N = pq is an odd composite number such that 3 ≤ p < q and kp ≤ q < (k + 1)p for some positive integer k; then p can be found in interval [e kl m , e kr m ] whose length is at most 1

⌋
, where e kl m = ⌋ and e b m = 2 m N − 1.
Proof The inequality kp ≤ q < (k + 1)p yields kp 2 ≤ N = pq < (k + 1)p 2 ; namely, Referring to the definition of the floor function and its property (P13) in (WANG X, 2017 Floor Function), it yields Corollary 1 Suppose N = pq is an odd composite number; then N can be factorized in at most 1 ⌋ steps of searches to find e p m that has a common divisor p with N.
Corollary 2 Suppose N = pq is an odd composite number and k ⌋ and e b m = 2 m N − 1.

Proof (Omitted)
Theorem 2 Suppose N > 1 is an odd integer and k 1 < k 2 are positive integers.Let ls k and By definition of the floor function, it holds ⌋ be the length of the odd interval I 0 = [e 0 m , e b m ], ∆ 1 , ∆ 2 and ∆ 3 , be respectively the lengths of I 1 , I 2 and I 3 ; then Proof Direct calculation by ( 5) yields ⌋ , which is the case of Lemma 6. Proof Referring to (6) it knows that ∆ k take two possible values given by k hold for arbitrary positive number k, it knows Taking in (10 node that are left to e b m+i , where m = ⌊ log 2 N ⌋ − 1 and i = 0, 1, ...; let odd interval [e n m+i , e b m+i ] contains n consecutive odd numbers; then the following statements hold.(1).e n m+i = e b m+i − 2(n − 1); (2).e p m+i lies in odd interval [e 0 m+i , e b m+i ] and there are p+1 2 nodes from e p m+i to e b m+i , where e 0 m+i = e b m+i − 2( illustrated in figure 1.

)
Figure 2. Locations of e p m (m > 2) 5) then by Lemma 1, ∆ k is the number of nodes contained in the odd interval [e kl m , e kr m ].By Lemma 5(1), it knows it holds e kl 1 m < e kr 1 m ≤ e kl 2 m < e kr 2 m where e kl 1 m = e b m − 2(lb k 1 − 1), e kl 2 m = e b m − 2(lb k 2 − 1), e kr 1 m = e b m − 2(ls k 1 − 1) and e kr 2 m = e b m − 2(ls k 2 − 1).Particularly, when k 2 = k 1 + 1 it holds e kl 1 m < e kr 1 m = e kl 2 m < e kr 2 m Proof Taking a = N in Lemma 6 immediately results in the theorem, which can be illustrated with figure 3. as illustrated in figure 3.

Figure 3 .m 3
Figure 3. Distribution of e kr 2 m , e kr 1 m , e kl 2 m and e kl 1 m