Some More New Properties of Consecutive Odd Numbers

The article proves several new properties of consecutive odd integers. The proved properties reveal divisors’ transition by subtracting two terms of an odd sequence, divisors’ stationary with adding or subtracting an item to the terms and pseudo-symmetric distribution of a divisor’s power in an odd sequence. The new properties are helpful for finding a divisor of an odd composite number in an odd sequence.


Introduction
Study of odd integers has been an important topic in number theory for several hundred years, as introduced in Dickson's book (Dickson, L. E., 1971).People have spent much time on studying the prime numbers, which are a special kind of odd integers, and obtained many excellent achievements as well as a lot of unsolved problems most of which are closely related with odd integers, as illustrated in Rosen's book (Rosen, K. H., 2011).Nowadays, the problem of factorizing a large odd number has still been a well-known difficult problem in the world, as Sarnaik S and Liu XX stated in their articles (Sarnaik, S., Gadekar, D., Gaikwad, U., 2014 ; Liu, X. X., Zou, X. X. & Tan, J. L., 2014), and Kessler overviewed in his book (Kessler G C. 2016).It is indubitable that, study of odd integers in different perspectives is helpful for knowing both the prime numbers and the factorization of integers.Based on such a point of view, WANG made studies on odd integers by several different approaches and obtained many new properties (WANG Xingbo, 2014-2017).Following the previous studies, this article aims at discovering some more new properties of consecutive odd numbers and intends to provide a mathematical foundation in people's knowing the distributions of odd integers' divisors.

Preliminaries
This section introduces symbols, definitions and lemmas that are necessary in later sections.

Symbols and Notations
Throughout this paper, an odd sequence is defined to be a sequence of odd numbers, e.g., 13,15,19,23,31.An odd interval [ , ] ab is a set of consecutive odd numbers that take a as their lower bound and b as their upper bound.For example, [3,11] {3,5,7,9,11}  .An odd interval [ , ] abis said to contain another odd interval [ , ]  cd , denoted by [ , ] [ , ]  c d a b  , if ,, abcand d satisfy one of the following three conditions (1) ac  and db  ; (2) ac  and db  ; (3) ac  and db  .Symbol ( , )  ab denotes the greatest common divisor of integer a and b.Symbol ( , , ) m n p  denotes the number of p's multiples from integer m to integer n.Symbol x   is to express x's floor function defined by , where x is a real number; and symbol x is to express x's odd floor function that is defined by , 1, x when x is odd x x when x is even

Lemmas
Lemma 1 (See in Rosen's book,2011) Let a, b, c and r be integers such that a bc r  ; then ( , ) ( , ) a b b r  . If m and n are odd integers and mn  , then ( , ) ( , ) m n n m n  Lemma 2 (See in WANG Xingbo's, 2014&2016) Let p be a positive odd integer; then among p consecutive positive odd integers there exists one and only one that can be divisible by p.Let q be a positive odd number and S be a finite set that is composed of consecutive odd integers; then S needs at least ( 1) 1 nq  elements to have n multiples of q.
Lemma 3. Let , mnand p be positive integers such that 1 p m n    ; then number of p's multiples from m to n is calculated by , ( , , ) . Grouped m integers Now group all the integers from 1 to n in the same way, as shown in figure 2.
. As a result, the total number ( , ) pn which says there are three 3's multiples that are produced by subtracting arbitrary two elements in S. In fact, it can see that the 3 multiples are 9 3 6,11 5 6     and 13 7 6 .
Theorem 3. Let p be an odd integer and n be a positive integer with np Proof.Since p and 1 a are odd integers, let 1 2 a p s  ; then the set * S can be equivalently rewritten by then terms in S are one-to-one mapping to the terms in * S and S contains n consecutive integers.By Lemma 3, the number of p's multiples in S is calculated by   , it knows that the first assertion of the corollary is sure to hold.(2 1)  (2) there are at most 1 s   consecutive odd integers from b to a or from c to a. Proof.First prove the case 21 ps   .Let ...,  is the remainder of 0 e divided by p and its value must be one of 1,3,…,p-2, the biggest of which then there must exist an odd integer b and an odd integer c such that (1) bac  and | It is known that, there are

3. Theorems and Proofs Theorem 1. Let n be a positive integer and
□

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Let p be an odd number and n be a positive integer with np  .Suppose 12 , ,..., Consequently, it knows that there are totally np  numbers that are of the form 2 p which is produced by   ; the nk  numbers are   , and when np   there are totally np   numbers that are of the form 2 p  which is produced by 2 be an odd integer and n be a positive integer with np  .Suppose ; then S contains at most one p's multiple and * S contains at most two p's multiples.When * S contains one p's multiple, it is either Given p is an odd integer bigger than 1, n is a positive integer with np  and □Corollary 2. Let p be an odd integer and n be a positive integer with np  .Suppose 12 { , ,..., } Similarly, it knows that, among s  consecutive odd integers next to a's right there must exist a c such that | sc  and there are at most 1 s   consecutive odd numbers either from c to a or from b to a.  consecutive odd numbers next to a's right there are odd numbers is c such that |  ), where s and t are odd integers bigger than 1;  ,  and  are positive integers that with 1   , 0   and 0   ; then among s  consecutive odd numbers next to a's left there are odd numbers is b such that | some way symmetrically distributed as divisors of odd numbers around a , as depicted in figure2.Figure 2. Symmetrically distributed powers of a's divisor , are in Corollary 7 Let a ps  be an odd composite integer, where p and s are odd numbers with 1 ps ; then there always exist an odd sequences 21 1 ...