Genetic Traits of Odd Numbers with Applications in Factorization of Integers

Transcription

1 Global Journal of Pure and Aled Matheatcs ISS Volue 3, uber (07, Research Inda Publcatons htt://wwwrublcatonco Genetc Trats of Odd ubers wth Alcatons n Factorzaton of Integers Xngbo Wang Deartent of Mechatroncs, Foshan Unversty, Foshan Cty, Guangdong Provnce, PRC, 58000, Chna Abstract The artcle roves that there exst genetc trats aong ntegers: an odd nuber wll regularly transt ts genes to other ntegers by akng tself be a dvsor of certan odd cooste nubers under defnte laws By the genetc trats, dstrbutve scoe of dvsors of an odd cooste nuber can be exactly known and lted n a defnte range by eans of valuated bnary tree Genetc structure, genetc grah and coleentary genetc grah are constructed n ter of the dscovered genetc laws ew aroaches for ralty test and nteger factorzaton are also ut forward wth nuercal exerents on factorzaton of soe Ferat nubers, Mersenne nubers and other bg ntegers Exerents ndcate that the new aroach s averagely faster than the Pollard Rho aroach Keywords: Integer factorzaton, Genetc law, Bnary tree, Algorth desgn MSC 000: A5,A05 I ITRODUCTIO Studyng ntegers by eans of bnary tree can reveal any new roertes of ntegers Artcle [] ut forward the concet of valuated bnary tree and roved soe fundaental laws on dvson relatons between the root and other nodes of an oddnuber-valuated tree Artcle [], followng the study of the artcle [], nvestgated several new roertes of odd nubers, ncludng laws of syetrc nodes, syetrc coon factors, subtrees dulcaton, subtrees transton, root dvson

2 494 Xngbo Wang and unfor su These roertes are called ausng roertes by artcle [] but n fact they are very serous and ortant for study of the odd nubers Ths artcle contnues revealng an ortant new roerty that dscloses a genetc trat of factors transtons aong odd nubers By the genetc trats, dstrbutve scoe of dvsors of an odd cooste nuber can be exactly known and lted n a defnte range that akes t easer to factorze an odd cooste nuber II PRELIMIARIES Defntons and otatons Ths artcle contnues adotng defntons and notatons related wth the valuated bnary tree and subtrees that were gven n [] and [] Odd nubers entoned n ths artcle are those bgger than If the root of a valuated bnary tree s 3, then the tree s called T 3 -tree, sly denoted by T3, as shown n fgure ote that each odd nuber bgger than ust be a node of T 3, hence odd nuber s usually wrtten by ts oston n T 3 For exale, s to ndcate the odd nuber s on the j th oston of the k th level n T 3, where k log ( k, j In dstngushng fro T 3, sybol T denotes a subtree whose root s (n T3 and sybol (, denote the node at the th oston on the th level n T ode (, and node (, geoetrcally syetrc on the th level thus they are called oston-syetrc nodes It s a conventon that any tree s root starts fro level 0 Sybol ( ab, or [ abn, ] ths whole artcle eans a set of consecutve odd nubers that are dstrbuted n the oen nterval ( ab, or the close nterval [ ab, ] 3 are k+ + k+ +3 k+ +5 k+ - Fgure T3 tree

3 Genetc Trats of Odd ubers wth Alcatons n Factorzaton of Integers 495 Leas Let (0,0 be an odd nuber and T be the (0,0-rooted bnary tree If (0,0 (0,0 3, then (0,0 (0,0 ( k j T (0,0 becoes T3 Let be a node n T (0,0 ; let (, be a node n T Relatonshs (0,0 aong (0,0 (0,0, T, (0,0, (0,0 ( k j (, ( k j and T are ntutonally dected by fgure (0,0 ( k j (0,0 ode n T3 (0,0 ode n T (0,0 (0,0 (, odes n T (0,0 (0,0 (, Fgure Relatonshs aong nodes of T3 tree, T3 s subtree and subsubtree Artcles [], [] and [3] have roven the followng Leas to 5 Lea For T(0,0 t holds ( There are k nodes on the k th level, 0,, k ; (0,0 ( ode s couted by (0,0 k k k (0,0 j ; k 0,,,; j 0,,, ( (0,0 (0,0 (3 Two oston-syetrc nodes, and (, (0,0 (0,0 (, (, (0,0 (,, satsfy ( (4 There s not a ultle of (0,0 before the level log (0,0, there are exactly ultles of (0,0 on the level log (0,0

4 496 Xngbo Wang Lea The th ( 0 level of subtree T ( k 0 s the ( k th level of T (0,0 and t (0,0 contans nodes ode of (0,0 (0,0 ( k, j (0,0 ( k j (, ( k j ( k j of T by the followng forula (3 (0,0 T (0 s corresondng to node (0,0 ( k j (0,0 (0,0 (, (, (, k k j ; j 0,,, ; 0,,; 0,,, (3 k j Lea 3 Two oston-syetrc nodes on each level of T (0,0 (, ft the followng laws (0,0 (0,0 (0,0 (, (, (, (, (, (0,0 (4 or (0,0 (0,0 (, (, (0,0 (, (, (5 or (0,0 (0,0 (0,0 ( k, j ( k, j (6 where 0 (0,0 Lea 4 (Syetrc Law of Coon Dvsors Suose node has a coon dvsor d wth (0,0, then d s also a coon dvsor of and (0,0 (, (0,0 (0,0 (0,0 ( k, j (0,0 ( k, j (, (, d gcd(, d gcd(, (0,0 (,, naely, Lea 5 Let be a ostve odd nteger; then aong consecutve ostve odd ntegers there exsts one and only one that can be dvsble by Let q be a ostve odd nuber and S be a fnte set that s coosed of consecutve odd nubers; then S needs at least ( n qeleents to have n ultles of q k k Lea 6 Suose s a odd nuber such that ( k, j and T an -rooted valuated bnary tree; then there are at least ultle-nodes of on level log of T for arbtrary nteger 0, and all these ultlenodes are subordnate to the syetrc law of coon dvsors Proof Frst, rove the followng assertons ( On the ( k th level of T, there are exact ultle-nodes of ; ( On the ( k th level of T, there at least nodes that are ultle-nodes of, where ; s

5 Genetc Trats of Odd ubers wth Alcatons n Factorzaton of Integers 497 (3 The ultle-nodes of are syetrcally dstrbuted on each of ther exstng levels In fact, there are k nodes on the ( k th level of T Take the case that k ( k, j, naely, takes ts axal value; then ownng to ( k k t knows by Lea 5 that, there are at least ultle-nodes on the ( k th level of T when The secal case when yelds k k ( ( k, j whch ndcates that there s at least ultle-node on the level k+ And the syetrc law ensures that there are exact, whch also valdates Lea (4 n another way ow snce k log ( k, j, t yelds Ths fnally valdates the lea ( k ( log 0 III MAI RESULTS AD PROOFS Man results nclude genetc trats of factors transtons aong odd nubers, buldng of genetc structure, genetc grah and coleentary genetc grah, dstrbuton of dvsors of an odd cooste nuber, and new crteron of ralty test and new aroaches to factorze an odd cooste nuber They are ntroduced searately n the followng sectons 3 Genetc Trats of Factors Transtons Theore (Genetc Law If node of T 3 can dvde (, of T also dvde (, k j (, (, and T of (, (, k j (, (, And t can also dvde nodes, then t can (, k j (, (,, (, k j (,, (, whose roots are (, and resectvely aely, (, transts ts genes to ts descendents by akng tself a dvsor of ts certan descendents

6 498 Xngbo Wang Proof The concluson that dvdng (, results n ts dvdng (, can be drectly obtaned by Lea to 4 ext s to show (, k j (, (, ( k, j ( k, j (, (, (, and In fact, let ; then by Lea t yelds (, whch says k j (, Then agan by Lea t holds (, (, k j (, ( k, j (, (, and (, k j (, ( k, j (, (, whch leads to ( k, j ( k, j (, (, (, and (, k j (, (, Theore (Genetc Law Let odd nuber (, be a ultlcaton of two odd nubers, and ( ls,, naely, (, ; then subtree ( l, s T(, nherts all genetc trats fro both and ( ls, In another word, f d(, s a coon dvsor of (, th, whch les at the oston on the th level n T, then d (, (, dvsor of (, and (, and s also a coon (, Proof Snce d(, s a coon dvsor of and ts descendant node k j (, hence d(, a and (, d(,, b, where a and b are ntegers bgger than Consequently d(, s of course a dvsor of (, because (, ( l, s (, ext s to show d (, (, k k Snce j and j, t yelds Rewrte d(, b by (, (, k j d a j, k (, d b j k (, ad(, ( l, s and let ; then t holds d b j k (, d b j d a k (, (,

7 Genetc Trats of Odd ubers wth Alcatons n Factorzaton of Integers 499 Hence Consequently t yelds ( d (, b a (, (, (, (, d ( b a ( l, s (, d ( a b a (, ( l, s (, whch says d(, s a coon dvsor of (, and (, Theore can be ntutonally dected by fgure 3 Fgures 4 and 5 are two exales of Theore, fgure 6 s an exale of Theore (k,j (, (, ( k, j (, (, (, k j (, (, (, k j (, (, (, k j (, (, Fgure 3 Gene Transtons between root and ts descendents

8 500 Xngbo Wang s ultles s ultles 3 s ultles Fgure 4 9 and 5 nhert 3 s genes and descend the to ther own descendents as 3 does s ultles s ultles 5 s ultles Fgure 5 35 and 45 nhert 5 s genes and descend the to ther own descendents as 5 does

9 Genetc Trats of Odd ubers wth Alcatons n Factorzaton of Integers 50 3 s ultles s ultles Fgure 6 5 nherts genetc trats fro both 3 s and 5 s 3 Genetc Grah of Pre uber Let > be a re nuber and T be the -rooted valuated bnary tree; then accordng to Theores and, transts ts genes to ts descendents, whch are actually nodes of T It can see that such heredty rocess s hghly related wth the logcal structure of the root and ts two nearest ultles n T as claed n Lea (4 Ths secton nvestgates such structure and the role t lays n the heredty rocess n T Defnton Let > be a re nuber and T be the -rooted valuated bnary tree; then the geoetrc structure fored by the root and ts two ultles on the level log of T together wth the aths fro to the two ultles s called genetc structure of, as llustrated n fgure 7 s genetc structure s denoted by sybol g( and ts fve eleents are denoted by g(0,0, g (,0, g (,, e (0,0 and e (0,, resectvely Level 0 Genetc Structure k,0 k,* Level log Fgure 7 Genetc Structure

10 50 Xngbo Wang Coents Snce there s a unque ath connectng the root and each of ts sons, aths are usually exressed wth sle straght lnes and ther concrete geoetrc shaes are gnored unless secal deands Defnton 3 Let > be a re nuber and T be the -rooted valuated bnary tree; then 's genetc grah G( s T's subtree that s recursvely generated by the followng rules G( s rooted by ; Each node n of G( has two suns, a left son and a rght son; the father and the two sons as well as the two aths connectng the father and the two sons resectvely for a genetc structure gn ( ; 3 Two dfferent nodes n and n satsfy g( n g( n ; and g n g n f n n ; 4 G( g( n n ( ( f and only Then by Defnton 3, Lea (4 and Lea, the followng Theore 3 and Theore 4 hold Theore 3 Let > be a re nuber and T be the -rooted valuated bnary tree; then 's genetc structure conssts of three nodes and two aths of T by ( (0,0 (0,0 g ; ( (0,0 (0,0 (,0 ( k, s, (, ( k, t g g, where log log k log, s ( /, t ( / ( e ath (0,0 g(,0 connects and e, and (0, g(, connects and Theore 4 Let > be a re nuber and T be the -rooted valuated bnary tree; then s genetc grah G( s a colete full bnary tree and can be recursvely constructed 33 Coleentary Genetc Grah of Pre uber It knows fro Defnton 3 that, each node of G( s a ultle of Snce there exst 's other ultle-nodes n T, t s andatory to defne the followng coleentary genetc grah to descrbe these nodes

11 Genetc Trats of Odd ubers wth Alcatons n Factorzaton of Integers 503 Defnton 4 Let > be a re nuber and T be the -rooted valuated bnary tree; then 's coleentary genetc grah G * ( s a bnary tree that s subordnate to the followng rules ( odes and edges of G * ( coe fro T and G * ( s rooted by ; ( Each node of G * ( s a ultlcaton of and an odd nuber bgger than ; (3 Arbtrary node * * ng ( such that n satsfes n G(, that s, Then by Lea 4, the followng Theore 5 holds G( G ( Theore 5 Let > be a re nuber and T be the -rooted valuated bnary tree; then G * ( s a syetrc bnary tree, that s, ts left subtree and rght subtree are subordnate to syetrc laws of oston and coon dvsors Fgure 8 shows the T3 tree, 3's genetc grah and ts coleentary genetc grah (a T3 tree (b 3's genetc grah (c 3's coleentary genetc grah Fgure 8 T3 tree, 3's genetc grah and ts coleentary genetc grah

12 504 Xngbo Wang Obvously, by defntons of G( and G * (, the followng Theore 6 holds Theore 6 Suose s an odd re nuber and T s the -rooted valuated bnary * tree; let kg be the level of T where level of G( occurs and k g be the level of T where level of G * * ( occurs; then k g k g 34 Genetc Laws of Factors Transton n Odd Cooste ubers Theore 7 Suose 3 q are odd nubers bgger and (, q ; let kq log q log q log q s ( q /, t ( q / and ; then there are at least (, ultle-nodes of that are syetrcally dstrbuted between and (, Proof Let t s yelds ( kq, s ( kq, t ; then by roertes of the floor functon, see n [4] and [5], t log q log q log q log q t s ( q / ( q / ( q / ( q / q (, whch says that there are at least q nodes between and (, Snce <q, t (, knows there ust exst at least one s ultle-node between and (, By syetrc law, the theore holds ( kq, s ( kq, t ( kq, s ( kq, t Theore 8 Suose (, s an odd nuber and (, q, where >0 s an nteger, 3 q are odd cored nubers; let log (, ; then there ust be at least two s ultle-nodes and two q s ultle-nodes on level of (, T All the ultle-nodes of and q are subordnate to the syetrc law and the s ultle-nodes are dstnct fro the q s ultle-nodes Proof The assuton that >0 and yelds (, (, (7 Snce (,, t knows (, and thus Because there are T and nodes on the level of (, for arbtrary 0 (,, t knows by Lea 5 that there are at least two s ultle nodes that are syetrcally dstrbuted on the level On the other hand, (, q and q 3 yeld (, 3q, naely,

13 Genetc Trats of Odd ubers wth Alcatons n Factorzaton of Integers 505 (, q (8 3 3 Hence there are at least two q s ultle-nodes that are syetrcally dstrbuted on the level By syetrc law, t s obvous that all the s and q s ultle-nodes are syetrcally dstrbuted ext s to rove that the s ultle-nodes are dstnct fro the q s ultle-nodes f and q are cored In fact, f a s ultle-node s also a q s ultle-node, or vce versa, then t ust be a ultle-node of q (, due to the coralty of to q Ths s contrast to the fact that (, has no ultle-nodes before level (, accordng to Lea (4 Hence the theore holds Corollary If > s an odd nuber and, then there are exactly least two s ultle-nodes that are syetrcally dstrbuted on level log (, of T (, (, Theore 9 Suose (, s an odd nuber and (, q, where > s an nteger, 3 q are odd core nubers; let log (, ; then there ust be at least two s ultle-nodes that are syetrcally dstrbuted on level of (, T Proof Snce log (,, there are nodes on level The nequalty 3 q yelds 3 (, q Hence yelds (, Referrng to nequalty (7, and t knows that when > Hence on the level, there s at least one s ultle-node By the syetrc law the level contans at least s ultle-nodes 35 ew Crteron of Pralty and Factorzaton of Integers Theore 0 Let (, be an odd nuber and T(, be the (, -rooted valuated bnary tree If (, has no coon dvsor wth any node fro level to level log (, of (, T, then (, s a re nuber Proof Use roof by contradcton Assue (, ( l, s to be a cooste nuber; then k and l By Lea (4 and Theore, ether or ( ls, has a dvsor after level and before level log (, of T (,, whch s contradct to the condton of the theore Hence the theore holds

14 506 Xngbo Wang Theore Let n and (,, where n,,, n are odd nubers bgger than ; then the bgger n s, the easer (, s factorzed If n and bgger s, the easer (, s factorzed Proof Let K log (, k log (,,, n K k ; then the log (,, s ( (, /, log (, t ( (, / and ; then by Lea 6 there are resectvely at least ultle-nodes of on level K of T(, By Theore 7, there ust exst (, (, ultle-nodes of n the nterval [ ( K, s, ( K, t ] that contans (, nodes of (, T Consequently, the bgger n s, the ore ultle-nodes are contaned n the nterval, and thus the easer (, s factorzed because each of the ultle-nodes has a coon dvsor wth (, ow consder the case n= By Lea 6, there are resectvely at least K k and Kk ultle-nodes of and on level K of T(, ote that, by Lea, the two (, nodes on level K, and (,, are the only ones that are ultles of both and ( Ks, ( Kt, Hence there are totally at least K k K k ultle-nodes of or on level K of (, T K k K k Let T( K, k, k ; then t yelds T K k k K k k k (,, ( Snce K k log log log log log log log log k k and T K k k, t results n (9 (,, log ( log Therefore, the bgger s, the ore ultles le on level K, and thus the easer (, s factorzed Theore Suose (, s an odd cooste nuber that fts (, and (, q such that and q are odd nubers such that 3 q ; let (, t log ( (, (, / ; use sybols (, ( K, ( q log (, K log, s ( (, / (, and ( K, ( resectvely to ndcate the (, frst q s and the frst s ultle-nodes that are left to the node K ; then there ( K,,

15 Genetc Trats of Odd ubers wth Alcatons n Factorzaton of Integers 507 (, are exactly nodes fro (, (, ( Ks, to K, there are at least ( K, ost (, nodes fro (, ( K, ( q to K, and there are at ost (, ( K, ( ( K, (, (, and at nodes fro (, to K, as llustrated by fgure 9, where the sybol Con eans counts of nodes ( K, Max range of q s ultles Max range of s ultles (, (, (, s (, ( q (, (, (, ( K (, Con ( / (, Con Con ( (, / Fgure 9 Dvsors dstrbuton (, Proof The frst concluson that there are exactly (, nodes fro to (, K ( K, can be drectly drawn fro Lea 5 ext s to rove the other ones Snce q and 3 q (,, t yelds (, and q (, Referrng to the roof of Theore, t knows that, when K log (, log (, and t ( (, / ( Ks, log (,, s ( (, /, there ust exst s and q s ultle-nodes that are (, (, (, syetrcally dstrbuted n nterval ( ( K, s, ( K, t on level K of T(, Let ( Kq, (, and ( Kq, be the two neghborng syetrc ultle-nodes of q; then there are q+ nodes between the two Snce q (, t yelds q (, (, whch says that there are at least (, (, nodes fro (, ( K, ( q to the node K ( K,

16 508 Xngbo Wang On the other hand, referrng to (8 yelds are both ostve ntegers, t yelds q Snce q and whch says there are at ost (, nodes fro (, ( K, ( q to K q (0 ( K, (, Slarly, let and (, be the s two neghborng syetrc ultle-nodes; ( K, ( K, then the nequalty (, results n (, (, Snce and (, are ntegers, t yelds (, ( whch says there are at ost (, (, nodes fro (, ( K, ( to the node K ( K, Theore 3 Let (, q be an odd cooste nuber such that and, where and q are odd core nubers that ft (, (, ; let sybols (, (,0 and be resectvely the leftost and the rghtost (, 3 q (, nodes on level + n the left branch of T(, ; let (, (, ( q and (, ( ndcate (, resectvely the frst q s and s ultle-nodes left to that s left to and (, (,, (, (, ( q be the node (, nodes away fro (,, and be the d-node (, (, that s rght to and (, nodes away fro (, (,0 ; then the dstrbuton of (, (, (, ( q, (,, (, (, (, ( q, (, ( llustrates (,0, (, and on level + s as fgure 0 (,

17 Genetc Trats of Odd ubers wth Alcatons n Factorzaton of Integers 509 Range of q s ultle node Range of q s ultle node (, (,0 (, (, (, ( q (, (, (, ( q (, (, ( (, (, (, Fgure 0 Dstrbuton of Crtcal odes (> Proof For convenence, denote the nuber (, by (, ( Then by Theore, t requres at ost ( (, nodes and at ost nodes to fnd a 's and a q's ultle-node fro the rghtost node on a level n the left branch of T(, Therefore, (, the node (, ( q s actually a boundary-ont that stos searchng a s ultle-node fro (, and starts searchng a q s ultle-node towards (, (, Snce t yelds when > (,0 (, ( (, ( ( Hence the nuber of nodes fro (, (, (, ( q to can never exceed the nuber of (, nodes n the left branch on level + of T (, because the later contans nodes By Theore 8, on level log (, there are at least 4 nodes that have coon dvsors wth (, It knows by the syetrc law that, aong nodes on level n the left branch of T(,, there are at least nodes that have coon dvsors wth (, (, By Theore, the two nodes, (, (, ( q and (, resectvely left to and rght to (, ( q (, (, do exst and they are

18 50 Xngbo Wang (, ow nvestgate the relatonsh between the d-node and the boundarynode (, (, (, ( q A drect calculaton shows and ( (, These two nequaltes ndcate the followng two conclusons ( (, ( When >, t always holds (, 0, whch eans that (, d-node, or t holds (, (, ( q s rght to the (, (, (, (, ( q (, (, [, ] (, ( The d-node (, s qute close to (, (, ( q (, (, ow t s u to nvestgatng the aounts of nodes n two ntervals (, (, and [ (, ( q, ] (, ote that ( (, (, ( ( [, ] (,0 (, ( q Snce ( when >, t knows that the nuber of nodes n (, (, (, (, the nterval [ (,0, (, ( q ] s bgger than that n the nterval [, ] Meanwhle, t can see that, when > t holds ( (, ( (, ( q (, (, (, (, whch eans [ (, (,0, (, ( q ] when > Suarzng all the cases dscussed above, t s sure the theore holds

19 Genetc Trats of Odd ubers wth Alcatons n Factorzaton of Integers 5 Theore 4 Let (, q be an odd cooste nuber such that and, where and q are odd nubers that ft 3 q; let (, (, sybols, (, (,0 (, and be resectvely the leftost, the ddle and the (, (, (, rghtost nodes on level n the left branch of T(, ; let (, ( ndcate the frst s (, ultle-node left to (, ; then (, (, (, (, ( s at ost (, nodes away fro (, (, (,, and t holds that [, ] f 4 and f 5 (, (, (, (, ( (, (, [, ] Proof Referrng to (, t yelds when 5 (, ( (,0 (, 3 ( (, ( (, 0 (, (, (, whch says [, ] f 5 (, ( (, (, The rest of the roof can refer to the roof of Theore 3 Corollary Let (, be an odd cooste nuber; then t requres at ost (, searchng stes to fnd a dvsor of (, Corollary 3 Let (, and k log (, wth 4 ; then (, s a re nuber f t has no dvsor n (, (, consecutve nodes left to (, Proof By Theore 4 and Corollary, the assuton that (, has no dvsor n (, than (, (, consecutve nodes left to eans that t has no dvsor that s less (,, whch eans (, s re Corollary 4 Let (, be an odd cooste nuber; then there exst aroaches that fnd a dvsor of (, n no ore than log (, searches Proof By genetc law, a dvsor d of (, les ether on (, s genetc structure or on ts coleentary genetc structure If d s on (, s genetc structure, by Theore 7

20 5 Xngbo Wang t takes at ost log (, stes to reach the level log (, along certan ath fro (, to the node that has d as a dvsor If d s on (, s coleentary genetc structure, t takes at ost log (, stes to the level after the level log (, because the level log (, dvsors by Lea 6 surely contans nodes that have d as ther 4 ALGORITHM DESIG AD UMERICAL EXPERIMETS Algorths to factorze odd cooste nubers can be desgned accordng to the revous theores Ths secton resents two basc algorths One s a sequental searchng (SS aroach based on Theore 4, the other s a subdvson and squeeze searchng (SSS aroach 4 Sequental Searchng Algorth Sequental searchng algorth searches a node of s ultles that contan coon dvsors wth the root, whch can reach O( n the best case and (0,0 n the worst case The algorth s as follows ======== Sequental Searchng Algorth========== Inut: Odd cooste nuber (0,0 Ste Calculate searchng level: K (0,0 log ; Ste Calculate the largest searchng stes: l ax ( (0,0 / ; (0,0 Ste 3 Calculate lower and uer lts: K ul, ll ul l ( K, ax ; Ste 4 Search n nterval [ ll, ul ] the frst odd nuber that has coon dvsor wth (0,0 ===============End of Algorth ============== 4 Subdvson & Squeeze Searchng Algorth (0,0 The sequental searchng algorth searches every ossble node fro K ul ( K, ax ll ul l Accordng to Theore t wll cost a lot of te when an odd cooste nuber contans only two factors that are very close to one another Usng to

21 Genetc Trats of Odd ubers wth Alcatons n Factorzaton of Integers 53 subdvson and squeeze search aroach can decrease the searchng stes ====== Subdvson Squeeze Searchng Algorth======= Inut: Odd cooste nuber (0,0, subdvson rato Ste Calculate searchng level: K (0,0 log ; Ste Calculate the largest searchng stes: l ax (0,0 Ste 3 Calculate varables: ( / ; (0,0 ul ; K ll ul l ; ( K, ax l ll l ax ; left l ; rght l Ste 4 If FndGCD((0,0,ll or FndGCD((0,0,ul Else or FndGCD((0,0,l return GCD; Begn loo ul ul ; ll ll ; left left ; rght rght If FndGCD((0,0,ll or FndGCD((0,0,ul or FndGCD((0,0,left or FndGCD((0,0,rght return GCD; End loo ===============End of Algorth ============== Coents The subdvson and squeeze searchng algorth can vary any dfferent seces when the subdvson rato vares For exale, the slest one s b-subdvson of the nterval [ ll, ul ]; the nterval [ ll, ul ] can of course be subdvded by other subdvsons For exale, subdvdng the nterval by the golden-rato s ore effcent to the cases that (0,0=q when q/ s close to the golden-rato Theoretcally, the ore sub-ntervals are obtaned, the faster the algorth works

22 54 Xngbo Wang 44 uercal Exerents uercal exerents are ade on a PC wth an Intel Xeon E5450 CPU and 4GB eory va C++ g bg nuber lbrary Exerent data orgnate fro two sources Soe are sall Mersenne and Ferat ubers; soe are taken fro artcles [6], [7] as well as art data n artcle [8] Excet for alyng the two aroaches ntroduced revously, Pollard Rho aroach s also adoted and rograed accordng the ntroducton n artcle [9] Tables and lst the exerental results It can see that the subdvson and squeeze aroach s averagely faster than the Pollard's Rho aroach, whch s averagely faster than the sequental aroach Table Exerents on Mersenne and Ferat ubers Sall Factor Searchng Stes Pollard's Rho Aroach Sequental Aroach Squeeze Aroach M67= M7= M83= M97= M03= M09= M3= F5= F6= F9= F0= F=

23 Genetc Trats of Odd ubers wth Alcatons n Factorzaton of Integers 55 Table Exerents on Soe Bg Integers s Factorzaton Searchng Stes Pollard's Rho Sequental Aroach Squeeze Aroach = = = = = = = = = = = = = = = = = = = = = = o result n a week o result n a week 56 = = = = = IV COCLUSIOS AD FUTURE WORK As stated n artcle [], uttng odd nubers on a bnary tree s a new aroach to study ntegers and t can derve odd nubers any new roertes Lke the results derved n ths artcle and n artcles [] and [], the new roertes do dsclose odd nubers any trats that have been rarely known before and are very useful n studyng and analyzng ntegers It can see fro ths artcle and the nuercal exerent that the new roertes of odd nubers can also rovde new aroaches to factorze ntegers It s sure that, cobned wth other knds of algorths, such as the algorths n artcles [7] and [8], the new aroach can reach an exected effcency And ths wll gve valuable gudance to future work

24 56 Xngbo Wang ACKOWLEDGEMETS The research work s suorted by the natonal Mnstry of scence and technology under roject 03GA78005, Deartent of Guangdong Scence and Technology under rojects 05A and 05A00040, Foshan Bureau of Scence and Technology under rojects 06AG003, Secal Innovatve Projects 04KTSCX56, 04SFKC30 and 04QTLXXM4 fro Guangdong Educaton Deartent The authors sncerely resent thanks to the all REFERECES [] WAG Xngbo, Valuated Bnary Tree: A ew Aroach n Study of Integers, Internatonal Journal of Scentfc and Innovatve Matheatcal Research (IJSIMR, 4(3, 63-67(06( DOI:0043/ [] Xngbo WAG, Ausng Proertes of Odd ubers Derved Fro Valuated Bnary Tree, IOSR Journal of Matheatcs, ( 6,VerV,53-57(06(DOI: 09790/ [3] WAG Xngbo, ew Constructve Aroach to Solve Probles of Integers' Dvsblty, Asan Journal of Fuzzy and Aled Matheatcs, (3,74-8(04 [4] Wang Xngbo, A ean-value forula for the floor functon on ntegers, Mathrobles Journal, (4,36-43(0 [5] WAG Xng-bo, Soe suleental roertes wth aendx alcatons of floor functon, Journal of Scence of Teachers College and Unversty (In Chnese,34(,8-9(04 [6] R Sheran Lahan, Factorng Large Integers, Matheatcs of Coutaton, 8(6, (974 [7] ShaohuaZhang, GonglangChen, ZhongngQn, et al, A Method of Factorng Large Integers,Inforaton Securty and Councaton rvacy, (7,08-09(005 [8] ZHAG Shu-e,SOG We-tang and SOG Wan-l, Dscusson on Mantssa ulthase artcle swar otzaton aled to large nteger factorzaton roble,couter Engneerng and Alcatons,46(5,05-08(00 [9] Sonal Sarnak, Dnesh Gadekarand Uesh Gakwad, An overvew to Integer factorzaton and RSA n Crytograhy,ITERATIOAL JOURAL FOR ADVACE RESEARCH I EGIEERIG AD TECHOLOGY,(9,-6,04

25 Genetc Trats of Odd ubers wth Alcatons n Factorzaton of Integers 57 [0] Xngbo WAG Seed and Seve of Odd Cooste ubers wth Alcatons In Factorzaton of Integers, OSR Journal of Matheatcs (IOSR-JM, (5, Ver VIII, 0-07(06 [] Xngbo WAG, Factorzaton of Large ubers va Factorzaton of Sall ubers, Global Journal of Pure and Aled Matheatcs, ( 6, (06

26 58 Xngbo Wang