IOSR Journal of Mathematcs (IOSR-JM) e-iss: 78-578, p-iss: 19-765X. Volume 1, Issue 6 Ver. V (ov. - Dec.016), PP 5-57 www.osrjournals.org Amusng Propertes of Odd umbers Derved From Valuated Bnary Tree Xngbo Wang 1 (Department of Mechatroncs, Foshan Unversty, Foshan Cty, Guangdong Provnce, PRC, 58000) Abstract: The artcle reveals some new propertes of odd numbers by means of valuated bnary tree. It frst systematcally nvestgates propertes of valuated bnary subtrees, ncludng calculaton law of nodes, symmetrc law of nodes ther common dvsors as well as duplcaton transton law of a subtree, then t dgs out several new propertes of the odd numbers that are bgger than 1. The new propertes clam that, n an odd valuated bnary subtree the sum of nodes on every level the sum of nodes n the frst fnte levels are multples of the root node, dvson of the sum of the nodes n the frst fnte levels by the root node remans the same value for all odd valuated bnary subtrees provded that the same number of levels are taken nto the sum. The new propertes can be valuable n studyng analyzng the securty key n nformaton system. Keywords: Odd numbers, Integer dvson, Multple MSC 000: 11A51,11A05 I. Introducton Artcle [1] put forward the concept of valuated bnary tree demonstrated that the valuated bnary tree could certanly dg out new propertes of ntegers. As follow-up work, ths artcle presents some other new amusng propertes of odd numbers n term of valuated bnary tree. II. Prelmnares.1 Defntons otatons A valuated bnary tree T, whch s defned n [1], s such a bnary tree that each of ts node s assgn a value. An odd-number rooted tree, whch s also defned n [1], s a recursvely constructed valuated bnary tree whose root s the odd number wth 1 1 beng the root s left rght sons, respectvely. For convenence, let symbol be the node on the j th poston of the k th level n T, where k 0 k 0 j 1; let symbol T denote a subtree whose root s symbol (, ) denote the node on the th poston of the th level n T.. Lemmas Let be an odd number T be the -rooted bnary tree, as shown n fgure 1. Then the followng lemma holds Lemma 1. Let T be a recursvely constructed -rooted bnary tree; then the followng statements hold (1) There are k nodes on the k th level, k 0,1,... ; () ode s computed by j 1; k 0,1,,...; j 0,1,..., 1 (1) k k k () The two nodes, (, ) (, 1 ) 1 (, ) (, 1 ), are at the symmetrc poston of the th level t holds () Proof. See n [1]. Lemma. For arbtrary m>0, Proof. m m m 1 ( 1)( 1) ( m 1) ( 1) m 1(mod ).. Snce the remnder of m dvded by s ether 1 or, t knows that one of m must be dvded by. Hence the lemma holds. DOI: 10.9790/578-10605557 www.osrjournals.org 5 Page
Amusng Propertes of Odd umbers Derved From Valuated Bnary Tree (1,0) (0,1) (,0) (,1) (,) (,) Fg.1. An -rooted bnary tree III. Man Results Proofs ew results nclude propertes of subtrees new trats of odd numbers. They are ntroduced separately n the followng two sectons..1 Propertes of Subtrees Theorem 1 (Law of odes Calculaton) The th ( 0 ) level of subtree T ( k 0 ) s the ( k ) th level of ( k. j) T, contanng nodes, the smallest bggest of whch are n term of T (, ) 1 k j, (, 1) 1 k j ode (, ) of T (0 1) s correspondng to node (, of T k j ) by the followng formula () ( k. j) (, ) (, ) (, ) k k j 1; j 0,1,..., 1; 0,1,...; 0,1,..., 1 () k j Proof. By Theorem n artcle [], the node at the th poston of the th level nt ( j+) th poston of the ( k ) th level n ( k, j ) k k T, namely, ( j ) 1 k k 1 j 1 ( j1) 1 k k 1 (, ) ( k, j ) DOI: 10.9790/578-10605557 www.osrjournals.org 54 Page by. By (1) t yelds s just the node at the Obvously, (, 0) 1s the smallest node k j (, 1) 1 s the bggest one. k j Theorem (Law of Symmetrc odes) odes on each level of T ( k>1) are symmetrc by the followng laws ( k, j ) ( k, j ) 1 ( k, j ) (4) or (, ) (, 1 ) ( k, j ) ( k, j ) 1 (, ) (, 1 ) (5) or 1 (6) ( k, j) ( k, j 1 ) where 0 1. Proof. By Theorem 1, (, ) 1 (, 1 ) ( 1 ) 1. Ths mmedately (, ) results n (5). Snce k j, (4) holds (6) can be valdated by (). (k,0) (k,1) (k,) (k,j) (k,*) Theorem (Law of Symmetrc Common Factor) Suppose node has a common factor d wth then d s also a common factor of Proof. (Omt). (, 1 ) (, ),
Amusng Propertes of Odd umbers Derved From Valuated Bnary Tree Theorem 4 (Law of Subtrees Duplcaton) Subtree T can be duplcated from T. Let (, ) be node of T (, ) be node of T ; then (, ) (, ) ( ), 0,,...; 0,1,..., 1 (7) Proof. By (1) t yelds 1 (, ) And by Theorem 1, (, ) ( k, j ) (, ) ( k, j) 1 1 ( ) 1 ( ) (, ), hence t yelds Take T 9 T as an example, the followng fgure llustrates the duplcaton law of subtrees. T 9 9 17 19 9= 0 (9-)+ 17= 1 (9-)+5 19= 1 (9-)+7 T 5 7 5 7 9 65 67 69 71 7 75 77 79 9 11 1 15 17 19 1 5 7 9 1 Fg.. T 9 duplcated from T Theorem 5 (Law of Subtrees Transton) Let T T ( ms, ) ( m k ) be two subtrees of T ; then t holds ( m, s) ( k, j ) (, ) ( m, s) (, ) ( ), 1,,...; 0,1,..., 1 (8) Proof. By Theorem 4, t holds ( ms, ) ( ), 1,,...; 0,1,..., 1 (, ) ( m, s) (, ) ( ms, ) (, ) ( m, s) (, ) ( ), 1,,...; 0,1,..., 1 Consequently t results n ( m, s) ( k, j ) ( ), 1,,...; 0,1,..., 1 (, ) ( m, s) (, ). ew Trats of Odd umbers Theorem 6 ( Law of Sum by Level) Sum of the th ( 0 ) level n Subtree T 1 0 ( k, j ) ( k, j ) (, ) Proof. By (7) symmetry of nodes t yelds 1 1 1 1 ( k, j ) ( k, j ) ( k, j ) 1 ( ) (, 1 ) (, ) (, ) 0 0 fts the followng dentty Theorem 7 (Law of Root Dvson) The root of subtree T can dvde the sum of the frst m (m0) levels of ( k, j ) ( k, j ) T, namely, can dvde (, ) (, ) m1 1. Or, 0 0 m1 1 0 0 (, ) s an nteger. Proof. By Theorem 6, sum of each level of T s a multple of. Hence the theorem holds. Theorem 8 (Law of Unform Sum) Arbtrary two subtrees, T T ( ms, ), satsfy (9) DOI: 10.9790/578-10605557 www.osrjournals.org 55 Page
Amusng Propertes of Odd umbers Derved From Valuated Bnary Tree ( m, s ) ( k, j ) (, ) (, ),,, 1,,...; 0,1,..., 1 ( m, s) Proof. By Theorem 7,, ( ms, ) (, ) ( ms, ) 1 m1 1 m1 m ( l, ) 0 0 0 m1 1 1 0 It yelds, 1 m (, ) ( m, s) ( m, s ) ( k, j ) ( l, ) ( l, ) m l, l, ( m, s) m 1 1, (, ) are two ntegers. ote that By Lemma, s an nteger, whch valdates the theorem. Take T 5, T 7 T 5 as examples, Theorem 7 8 can be llustrated ntutonally by followng fgure. (10) 5 1 (, ) 0 0 45 45 85 5 9 11 17 19 1 5 7 9 41 4 45 47 7 1 (, ) 0 0 595 595 85 7 1 15 5 7 9 1 49 51 5 55 57 59 61 6 5 1 975 (, ) 975 85 0 0 5 69 71 17 19 141 14 7 75 77 79 81 8 85 87 Fg.. Root dvson unform sum n subtrees IV. Conclusons As stated n artcle [1], puttng odd numbers on a bnary tree s a new approach to study ntegers t can derve many new propertes of the odd numbers. Lke the results derved n ths artcle n artcle [1], the new propertes can be helpful to underst more trats of odd numbers mght be useful n studyng DOI: 10.9790/578-10605557 www.osrjournals.org 56 Page
Amusng Propertes of Odd umbers Derved From Valuated Bnary Tree analyzng securty keys. In addton, bnary tree s excellent behavor n parallel computng s sure to make the new approach to obtan brllant achevements n future work. Acknowledgements The research work s supported by the natonal Mnstry of scence technology under project 01GA78005, Department of Guangdong Scence Technology under projects 015A00401105 015A010104011, Foshan Bureau of Scence Technology under projects 016AG10011, Specal Innovatve Projects 014KTSCX156, 014SFKC0 014QTLXXM4 from Guangdong Educaton Department. The authors sncerely present thanks to them all. References [1] WAG Xngbo. Valuated Bnary Tree: A ew Approach n Study of Integers, Internatonal Journal of Scentfc Innovatve Mathematcal Research (IJSIMR), Volume 4, Issue, March 016, PP 6-67 [] WAG Xngbo. Analytc Formulas for Computng LCA Path n Complete Bnary Trees, Internatonal Journal of Scentfc Innovatve Mathematcal Research (IJSIMR) DOI: 10.9790/578-10605557 www.osrjournals.org 57 Page