Internatonal Journal of Scentfc Innovatve Mathematcal Research (IJSIMR) Volume 4, Issue 3, March 6, PP 63-67 ISS 347-37X (Prnt) & ISS 347-34 (Onlne) wwwarcournalsorg Valuated Bnary Tree: A ew Approach n Study of Integers WAG Xngbo Department of Mechatroncs Foshan Unversty, Foshan Cty, Guangdong Provnce, PRC wxbmal@msncom Abstract: The artcle frst puts forward the concept of valuated bnary tree; then maes a study on factors multples among the nodes of a bnary tree that s valuated wth odd numbers obtans several new results that are valuable n study of nteger dvson The artcle also demonstrates that the approach of valuated bnary tree s ndeed practcal effectual because all ts proofs deductons are very elementary ntutve, whch s easer to underst utlze Keywords: Bnary tree, Valuated, Integer dvson, multple ITRODUCTIO Bnary tree has been a famlar term n schoolboos Study of ths non-lnear data structure was a long tme ago no one n present day thns t s worth to have a study on the bnary tree Compared to the fashonable bonformatcs or quantum computaton, the bnary tree s ndeed an old thng However f t s combned wth an ever-older thng, t reveals younger stronger trats Wth valuated bnary tree, I obtan many new results when I study the old problems n elementary number theory Hence I ntroduce the valuated bnary tree n ths artcle use t to prove some theorems that dsclose new propertes of nteger dvson Readers can see that the proofs deductons are very elementary ntutve t s easer to underst PRELIMIARIES Ths secton gves defntons, notatons lemmas that are needed n later secton Also the studed queston s presented n ths secton Defntons, otatons Queston Defnton The floor functon of a real number x s denoted by x, t fts x x x or equvalently x x x The bblography [] lsts propertes of the floor functon Hence here I omt the detals Defnton A valuated bnary tree s such a bnary tree that each of ts node s assgn a value If all the nodes are assgned an nteger number, then t s called an nteger-valuated bnary tree We use symbol T to denote a bnary tree Defnton 3 Assgn the root of a full bnary tree wth an odd number, assgn the left the rght sons of the root wth respectvely, assgn all the other nodes of the tree recursvely by the prevous regulaton Ths wll obtan an -rooted tree For example, fgure demonstrates a 7-rooted tree ARC Page 63
WAG Xngbo 7 3 5 5 7 9 3 Fg A 7-rooted bnary tree Queston Let us construct an -rooted bnary tree, where s an odd number bgger than one For convenence, we denoted the root by the node on the th poston of the th level by,, where, as shown n fgure Then under what condton can dvde the other nodes?,,,,,,3,,,,,* Lemmas Fg An -rooted bnary tree Lemma Let p be a postve odd nteger; then among p consecutve postve odd ntegers there exsts one only one that can be dvsble by p Proof See n [] Lemma Let be an nteger; then log log ( ) log Proof See n [3] Lemma 3 Let q be a postve odd number, S { a Z } be a set that s composed of consecutve odd numbers; f a Ss a multple of q, then so t s wth a q Proof Let a q ( ) ; then t yelds a a a q q ( ) q q q q 4 () Lemma 4 Let q be a postve odd number S be a fnte set that s composed of consecutve odd numbers; then S needs at least ( n ) q elements to have n multples of q Proof Let S { a, a,, am } be the fnte set that contans n multples of q Obvously, a s the smallest one am s the bggest one n the set ow consder the frst multple of q If a s the Internatonal Journal of Scentfc Innovatve Mathematcal Research (IJSIMR) Page 64
Valuated Bnary Tree: A ew Approach n Study of Integers frst one, then S needs at least ( n ) q elements to have n multples of q, as shown n fgure 3 Otherwse t must hold m ( n ) q a, a,, a, a,, a,, a,, a, a,, a m q q q ( n ) q nq nq m q q q m nq 3 MAI RESULTS AD PROOFS Fg3 n multples wth a beng the frst one Theorem Let T be -rooted bnary tree Then the followng statements hold () There s not a multple of before the level log () There are exactly multples of on the level log ; there are at least multples of on each level after the level log (3) There are at least nodes that are not multple of on each level from after the level log (4) If, have a common factor d, then factor, where,, also have d to be ther common (5) On the same level, there s not a node that s a multple of another one Proof By defnton 3, t s easy to obtan T s followng propertes () There are nodes on the th level () ode, s computed by,,,;,,, () (3) The two nodes,,,, are at the symmetrc poston of the th level t holds (3),, To analyses relatonshp between,, we rewrte () by (, ),,,;,,,, (4), Obvously, for a specfed level, say the th level, (, ) s a functon of Table lsts change of the values of (, ) It can see that, when {, 3,, ( )}, totally,,, ( ), (, ) numbers; when, ( ), (, ) traverses {, 3,, ( )}, totally numbers traverses the whole set,,,, (, ) {, 3,, ( )},, {,,, } (, ) {, 3,, ( )},, {,,, } (5) Internatonal Journal of Scentfc Innovatve Mathematcal Research (IJSIMR) Page 65
WAG Xngbo Table Change of values of (, ), (, ) ( ) ( ) ( 3) ( 3) 4 ( 5) ( 5) ( ) 3-3 ( ) - + ( ) 3 +3 ( ) 5 +5 ( ) 3 ( 3) ( ) ( ) ow t nows from (4) (5) that can not dvde, when or log ( ) By Lemma, we now log ( ) log ; hence when log or log, can not dvde, That s what statement () says log log ow consder the case log Ths tme t holds, namely or Snce s an odd number, t must be one of the element n the set S { 3,, ( )} amely, there exsts a {,,, } such that (, ) t consequently holds, by (3) ote that + s not an element of the set S,, t shows that there must be a * {,,, } such that (, *), namely, * cannot be multples of Ths valdates statement (3), * ow we prove that on the level log there exact multples of In fact, from log log log we now Internatonal Journal of Scentfc Innovatve Mathematcal Research (IJSIMR) Page 66, whch says that the nodes on the level log do not exceed Therefore there cannot be more than multples on the level accordng to Lemma 4 Hence the statement () s sure Statement (4) naturally can be drectly proved true by (4) Tae an arbtrary level, say the th level ote that the smallest the bggest node on the level are ( ( ) ) ( ) ( ) respectvely,, t ( ) ( ) mmedately nows that no node can be a multple of the other one on the level Ths s what the statement (5) says And by now the whole theorem s proved to be true 4 COCLUSIO It s of course a new approach to study ntegers wth a bnary tree The research n ths artcle shows that the new approach s both elementary ntutve, s better n dggng out new propertes of ntegers Actually, I have dug out many other propertes of ntegers I wll publsh them n later artcles Ths artcle s the frst one of the new approach, the purpose of ths artcle s to ntroduce the dea the thought of the approach I hope t can be concerned better achevements can be obtaned
Valuated Bnary Tree: A ew Approach n Study of Integers ACKOWLEDGEMETS The research wor s supported by the natonal Mnstry of scence technology under proect 3GA785, Department of Guangdong Scence Technology under proects 5A345 5A4,, Foshan Bureau of Scence Technology under proects 3AG7, Specal Innovatve Proects 4KTSCX56, 4SFKC3 4QTLXXM4 from Guangdong Educaton Department The authors sncerely present thans to them all REFERECES [] Graham R L, Knuth D E Patashn O "Integer Functons" Ch 3 n Concrete Mathematcs: A Foundaton for Computer Scence(nd ed),addson-wesley Professonal;pp 67- [] WAG Xngbo ew Constructve Approach to Solve Problems of Integers' Dvsblty, Asan Journal of Fuzzy Appled Mathematcs,4, (3):74-8 [3] WAG XngboSome Supplemental Propertes wth appendx Applcatons of Floor Functon,Journal of Scence of Teachers' College Unversty(n Chnese),4,34(3):7-9 AUTHOR S BIOGRAPHY WAG Xngbo, was born n Hube, Chna He got hs Master Doctor s degree at atonal Unversty of Defense Technology of Chna had been a staff n charge of researchng developng CAD/CAM/C technologes n the unversty Snce, he has been a professor n Foshan Unversty, stll n charge of researchng developng CAD/CAM/C technologes Wang has publshed 8 boos, over 8 papers obtaned more than patents n mechancal engneerng Internatonal Journal of Scentfc Innovatve Mathematcal Research (IJSIMR) Page 67