PRELIMINARIES This section lists for later sections the necessary preliminaries, which include definitions, notations and lemmas.

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1 MORE O SQUARE AD SQUARE ROOT OF A ODE O T TREE Xingbo Wang Departent of Mechatronic Engineering, Foshan University, PRC Guangdong Engineering Center of Inforation Security for Intelligent Manufacturing Syste, PRC State Key Laboratory of Matheatical Engineering and Advanced Coputing, PRC ABSTRACT: The article proves several iportant properties of the square and the square root of a node of T tree The new properties describe how the square and the square root of a node are distributed on the T tree and are helpful to locate divisors of a coposite integer KEYWORDS: Square, Square Root, Integer, Binary Tree ITRODUCTIO The square and the square root are undoubtedly very iportant operations for a nuber As a new structure of odd integers, the T tree, which was introduced in WAG (06 & 08), is of course necessary to ae clear these two operations In fact, for a given node in the tree, the proble where its square and its square root locate is surely a fundaental proble Soe general properties of the square of a node in T were entioned in WAG (08) and CHE (08) disclosed several properties of the square root of a node However, one can see that, there are still a lot of unnown properties This paper shows a little ore of the properties related with the square and the square root of a node PRELIMIARIES This section lists for later sections the necessary preliinaries, which include definitions, notations and leas Definitions and otations Let S be a set of finite positive integers with s 0 and sn being the sallest and the biggest nodes respectively; an integer x is said to be claped in S if s 0 x s n Sybol x S indicates that x is claped in S Sybol x is the floor function, an integer function of real nuber x that satisfies inequality x < x x, or equivalently x x< x+ In this whole paper, sybol T is the T tree that was introduced in WAG (06 & 08) and sybol (, j) is by default the node at position j on level of T, where 0 and 0 j By using the asteris wildcard *, sybol (,*) eans a node lying on level An + + integer X is said to be claped on level of T if X and sybol X indicates X is claped on level If a positive integer X is claped on level and there is a node Y of T satisfying X = Y, then X is said to be a floor square root of the node Y and Y is called a square source of X Print ISS: 05-9, Online ISS: ISS 05-0

2 Rear CHE (08) put forward the concept that an integer is claped on a level of T In CHE s paper, a positive integer X was said to be claped on level of T if X Since + is the rightost node on level and + + is the leftost node on level, there is an integer + between the two In order to avoid leaving out the nuber + + +, this paper redefines it by X Leas Lea (See in WAG (08)) T Tree has the following fundaental properties (P) Every node is an odd integer and every odd integer bigger than ust be on the T tree Odd integer with > lies on level log (P) On level with = 0,,, there are nodes starting by + + and ending by + + naely, + with j = 0,,, j (, ) [, ] +, (P) (, j) is calculated by + (, j) = + + j, j = 0,,, (P4) Multiplication of arbitrary two nodes of T, say (, α ) and ( n, β ), is a third node of T Let n J = (+ β ) + (+ α) + αβ + α + β ; the ultiplication (, α ) ( n, β ) is given by J + n+ (, α) ( n, β) = + + If + J n + <, then (, α ) ( n, β ) = ( + n +, J) lies on level + n+ of T ; whereas, if + J n + = with + χ J n + = lies on level + n+ of T (, α ) ( n, β ) ( + n+, χ ) (P5) Product = is a left node of T, and it lies on level + or + (, α ) (, α) (, α) Lea (See in WAG (07)) For real nubers x, y and positive integer i, it holds (P) x y x y i (P) i ; x < n x < n i, where n is an integer, MAI RESULTS AD PROOFS Theore Let be a positive integer; then there are + consecutive integers n n n + that satisfy n =,,, i ( i=,,,+ ) Proof Consider an arbitrary integer n such that n = ; then by definition of the floor function it holds n < + That is n< + + Print ISS: 05-9, Online ISS: ISS 05-0

3 Hence the + integers, n satisfying n = i ( i=,,,+ ) = +, n, n, 0 = + = +, and n = +, are the integers + Proposition Let (, α ) be a node of T with > 0 ; then when (, ) α α lies on level + ; otherwise it lies on level + Particularly, = + and (, ) (+, ) = + + Proof Direct calculation shows (, α ) = ( + α + ) = + (α + α + + α) + (,0) (+, ) By Lea (P4 & P5), it nows that α lies on level + if and only if J α α α = < Consequently, (, ), + α + α + + α < + α + ( + ) α + < ( + ) + ( + ) 4 ( ) 0 α < 0 α < ( + ) α < + + 0α which validates the first part of the proposition The second part is easily obtained by the following calculations = ( ) = + + = (,0) (+, ) = ( + ( ) + ) = ( + ) (, ) = = = + + = ( ) + + (+, ) Rear The condition > 0 in Proposition is proposed because it can get rid of the case = = 9=, which is the unique exaple that violates = + (0,0) (,0) (,0) (+, ) Print ISS: 05-9, Online ISS: ISS 05-0

4 Proposition Let be a positive integer, (+,*) and ( +,*) be nodes of T ; then + ( (+,*) ) and + ( (+,*) ) On level there is not a node (,*) satisfying (,*) level + Proof (,*) and there is neither a node (+,*) satisfying (+,*) on + and ( +,*) being the nodes on levels + and + respectively yields and ote that Hence it holds and 4 + < < +, it yields + That is ( (+,*) ) and + Considering the biggest node < + < (+,*) + = + + (+,*) < + < + + (+,*) < (+,*) + + = ( ( ) (,*) + < (+,*) (, ) (+,*) ( ) on level, it yields = < < (,*) (,*) (, ) (, ) Liewise, considering the sallest node ( +,0) on level +, it yields (+,0) = + > (+,*) (+,0) > Hence there is not a node (,*) satisfying (,*) ( +,*) satisfying (,*) +, and there is neither a node Print ISS: 05-9, Online ISS: ISS

5 Proposition ode (, j) ( > ) of T satisfies Or equivalently (, j) < (, j ) < Proof Since +, it yields < < ; hence it holds (, j ) (, j ) + + < (, j ) < By Lea (P), the inequality holds + + yields + + and + + +, hence it < (, j ) < By Lea (P) it iediately leads to which is + (, j) ( = ) (, j ) < Rear This proposition is a odification of the Corollary in CHE (08) That paper claied that + (, j) or (, j) However, seeing fro Propositions, and, one can see that + (, j) never occurs log Theore Given > be an odd integer; then Proof Let = log By Lea (P), is a node on level of T By Proposition it nows log, that is Exaple Table lists several odd integers that are randoly piced, and their positions log in T as well as their square roots in T It can see that, holds for each nuber Readers can chec it anually or with Matheatica Print ISS: 05-9, Online ISS: ISS

6 Table Odd Integers and their square roots in T Odd Integer s int 57 (9,46) 049 (,05) (7,606) (,7664) (5,878507) log & its level 4 57 = = = = = 0 Rear Table can be easily checed anually or with Matheatica When prograed as follows, f[x_]:=floor[floor[log[x]/log[]]/]-; g[x_]:=floor[log[sqrt[x]]/log[]]-; r[x_]:=floor[sqrt[x]]; indata={57,049,86757,69475, }; r=table[f[indata[[i]]],{i,5}]; r=table[g[indata[[i]]],{i,5}]; r=table[r[indata[[i]]],{i,5}]; t={r,r,r}//matrixfor the screenshot in Matheatica 70 is as figure Figure Screenshot of the progra and outputs Print ISS: 05-9, Online ISS: ISS

7 Corollary Let > be an odd integer and = log ; then when is odd whereas Proof By Theore, = l+, + ( ) when is even + ( ) is sure ow consider the fact that, whether = l+ or + + = l + always holds By Proposition, it nows that, ( ) when is odd whereas + ( ) when is even COCLUSIO The square and the square root are iportant nubers For an integer, the square root is essentially iportant because it is the cut-off point of divisors of a coposite integer Study of these nubers is helpful to understand distribution of the divisors on T The Theore proved in this paper is of course a foundation for us to now where the square root of a node lies Hope it is helpful in the future Acnowledgeent The research wor is supported by the State Key Laboratory of Matheatical Engineering and Advanced Coputing under Open Project Progra o07a0, Departent of Guangdong Science and Technology under project 05A00040, Foshan Bureau of Science and Technology under projects 06AG00, Project gg04098 fro Foshan University The author sincerely present thans to the all REFERECES CHE G, LI J (08) Brief Investigation on Square Root of a ode of T Tree, Advances in Pure Matheatics,8(7) WAG X (06) Valuated Binary Tree: A ew Approach in Study of Integers International Journal of Scientific and Innovative Matheatical Research, 4() 6-67 WAG X (07) Brief Suary of Frequently-Used Properties of the Floor Function, IOSR Journal of Matheatics, (5) WAG X (08) T Tree and Its Traits in Understanding Integers, Advances in Pure Matheatics, 8(5) Print ISS: 05-9, Online ISS: ISS