Seed and Sieve of Odd Composite Numbers with Applications in Factorization of Integers

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1 IOSR Joural of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume 1, Issue 5 Ver. VIII (Sep. - Oct.01), PP Seed ad Sieve of Odd Composite Numbers with Applicatios i Factorizatio of Itegers Xigbo WANG 1 (Departmet of Mechatroics, Fosha Uiversity, Fosha City, Guagdog Provice, PRC, 58000) Abstract: The article aims at owig the type of factors by which a multiplicatio of itegers is produced. By puttig forward the cocept of composite seed of a odd composite umber, the article costructs a sieve to select ad produce odd composite umbers of the form 1 ad the demostrates several theorems related with factorizatio of a odd composite umber i terms of its seed. It shows that factorizatio of a big odd umber ca be coverted ito that of small umbers icorporated with the big umber s seed. The article also maes a ivestigatio o ew characteristics of odd umbers as well as their factorizatio ad reveals several iate laws that are helpful for fidig factors of odd umbers. The revealed laws idicate that a odd umber might fid its factors ear the factors of its eighborig odd composite umbers. Subtle mathematical deductios are give for all the coclusios ad ecessary examples are preseted with detail iterprets. Keywords: Sieve, Odd umbers, Iteger Factorizatio, Iformatio security I. Itroductio Positive odd umbers that are of the form with iteger are geerally classified ito two ids, odd prime umbers ad odd composite umbers. The study of these umbers has bee topics i umber theory for several hudred years, as itroduced i [1]. People have spet much time o study of the prime umbers ad factorizatio of the composite umbers, ad the problem of factorizig a large odd umber has still bee a wellow difficult problem i the world, as stated i articles [] ad [3]. However, whe aalyzig the preset literatures, oe will see that, few literatures cocer the iate costituets of a multiplicatio that comes from prime factors multiplyig. A recet study o multiplicatio of two odes of the valuated biary tree, which is itroduced i articles [4], implicates the problem of owig the multiplicatio s sources, that is, by what id of odes the multiplicatio is produced. This problem is alteratively the problem of owig what id of prime factors a composite umber is produced by. For example, 45=59 with 5 ad 9 beig of the form 4+1, 1=37 with 3 ad 7 beig of the form 4-1 ad 15=35 with 3 ad 5 beig 4-1 ad 4+1 respectively. It is sure that, owig the factors sources of a composite umber is helpful to reduce a large amout of computatios whe searchig a factor of the composite umber. This article aims at solvig the problem. The article first puts forward the cocept of composite seed of a odd composite umber, the costructs a sieve to select ad produce odd composite umbers, ad the proves several theorems related with factorizatio of a odd composite i terms of the seeds. Also the article maes a ivestigatio o characteristics of odd umbers of the form 1 together with their factorizatio ad obtais several ew properties that are helpful for factorizatio of a odd composite umber. II. Prelimiaries.1 Defiitios ad Notatios I this whole article, symbol Z deotes all itegers, symbol Z deotes all the positive itegers ad 0 symbol Z deotes the positive itegers together with 0. Symbol AB meas that result B is derived from coditio A; symbol x Z meas there exists a x i set Z +. Symbol a b meas b ca be divided by a; symbol a Πb meas b caot be divided by a. Symbol (a, b) ad [a, b] are to express respectively the greatest commo divisor (GCD) ad the least commo multiple (LCM) of itegers a ad b. I this whole article, odd umber refers to that is bigger tha 3 uless special remar is made. Iteger of 3 s multiple is simply deoted by 3_m. Defiitio 1: A iteger is called a composite seed, or simply a seed, of umber +1(or -1) if the umber +1(or -1) is a odd composite umber; meawhile, the composite +1(or -1) umber is a sprout of the seed. For example, 8 is the seed of 49 ad 11 is the seed of 5; 49 ad 5 are sprouts of 8 ad 11 respectively.. Lemmas Lemma 1: A arbitrary o-egative itegers N 0 ca be expressed by N r,0 r 5, ad a odd umber bigger tha 3 must be of the form either N 1 or N 1, where Z. DOI: / Page

2 Seed ad Sieve of Odd Composite Numbers with Applicatios I Factorizatio of Itegers Lemma [5]: A odd umber of the form of 3 1must be also of the form of 1; a odd umber of the form of 3 must be also of the form of 1. A composite umber 1must be product of either two factors of the form 1or two factors of the form 1; a composite umber 1must be product of oe factor of the form 1with the other of the form 1. Lemma 3[]: Let p be a positive odd iteger; the amog p cosecutive positive odd itegers there exists oe ad oly oe that ca be divisible by p. Lemma 4[4]: Let q be a positive odd umber, S { a i i Z be a set that is composed of cosecutive odd umbers; if a S is a multiple of q, the so it is with a q. III. Mai Results ad Proofs Mai results i this sectio iclude 3 parts. The first part shows the critical pole of the seed i factorizatio of odd composite umber 1, the secod costructs a sieve to produce composite umbers of 1 ad shows the way to apply the sieve to factorize composite umbers of 1, the third is to show some ew properties of the umbers 1 ad their aids to itegers factorizatio. 3.1 Seeds ad Factorizatio of Odd Numbers Theorem 1: The seed of a odd composite umber is highly related to the factorizatio of the composite umber by the followig rules. (i) The ecessary ad sufficiet coditio for a composite umber 1 has a factor x 1 or y 1is that (x 1) ( x) or (y 1) ( y) respectively. (ii) The ecessary ad sufficiet coditio for a composite umber 1 has a factor x 1 or y 1is that (x 1) ( x) or (y 1) ( y) respectively. Proof: Here the proof is oly for (i) sice (ii) ca be proved by the same meas. Sufficiecy x 1 ( x) (x 1) m x 1 3xm x m 1 (x 1)(m 1) y 1 ( y) (y 1) m y 1 3ym y m 1 (y 1)(m 1) Necessity (x 1)(m 1) 3mx x m 1 ( m(x 1) x) (y 1)(m 1) 3my y m 1 ( m(y 1) y) 1 m(x 1) x (x 1) ( x) m(y 1) y (y 1) ( y) The followig Theorem 1 ca alteratively state theorem 1*. Theorem 1*: All the odd umbers have the followig properties. (i) The ecessary ad sufficiet coditio for a composite umber 1 ca be factorized by 1 (l1 1)(1 1) or 1 (l 1)( 1) is that l1 (1 1) 1 1(l1 1) l1 or l( 1) (l 1) l, where l 1, l, 1 ad are itegers less tha ; (ii) The ecessary ad sufficiet coditio for a composite umber 1 ca be factorized by 1 (l 1)( 1) is that l( 1) (l 1) l, where lare, positive itegers less tha ; 3. Sieves of Odd Composite Numbers Costruct four sequeces, S1, S, S1 ad S, such that S1 { N 1 ( 1)( l 1);, l Z S { N 1 ( 1)( l 1);, l Z S1 { N 1 ( 1)( l 1);, l Z S { N 1 ( 1)( l 1);, l Z Obviously, each item i S 1, S, S1 ad S is a seed of a odd composite umber. By Theorem 1*,, S1 S, S1 ad S ca be redefied by S1 { l( 1) (l 1) l;, l Z (1) S { l( 1) (l 1) l;, l Z () S1 { l( 1) (l 1) l;, l Z (3) S { l( 1) (l 1) l;, l Z (4) DOI: / Page

3 Seed ad Sieve of Odd Composite Numbers with Applicatios I Factorizatio of Itegers Note that, for a fixed, S1 is a arithmetic progressio with l l beig the iitial term ad l 1beig the commo differece i terms of coutig from l; hece S1 ca be equivaletly expressed by S 1 { s 1, s 1, l s0 ( 1) d, s0 l l, d l 1;, l Z (5) Actually, let K l 1 the s 1, l l ( 1)( l 1) l l l l 1 l(( l 1) 1) l 1 l(k 1) K K(l 1) l which asserts (4) is equivalet to (1). Similarly, S ad S ca be equivaletly expressed by S { s s s ( 1) d, s l l, d l 1;, l Z (), l, l 0 0 S Z (7) 1 { s 1, l s 1, l s0 ( 1) d, s0 l, d l 1;, l S Z (8) { s, l s, l s0 ( 1) d, s0 l, d l 1;, l Sequeces (1), (), (3) ad (4) or their equivalet expressios (5), (), (7) ad (8) ca of course form a sieve to select every odd composite umber of 1 with Z. I additio, the followig Theorem ad 3 provide computatioal foudatios. Theorem : Suppose N is a odd composite umber; the the followig statemets are true. (i) If N 1 ad there exists a (ii) If N 1 ad there exists a (iii) If N 1ad there exists a (iv) If N 1ad there exists a 1 such that ( 1) ( ), the ( 1) N. 1 such that ( 1) ( ), the ( 1) N. 1 such that ( 1) ( ), the ( 1) N. 1 such that ( 1) ( ), the ( 1) N. Proof: Tae the cases (i) ad (ii) as examples. It ca see that ( 1) ( ) a( 1), a Z 1 ( a( 1) ) 1 ( 1)(a 1) which validates ( 1) N. Now sice 1is a factor of N 1, it leads to 1 1 ( 1) 1 1 ( 1) 1 l l 1 which fiishes proof of the case (i). For case (ii), it ( 1) ( ) a( 1), a Z 1 ( a( 1) ) 1 ( 1)(a 1) ad ( 1) 1 l l Theorem 3: Suppose N is a odd composite umber; the the followig statemets are true. (i) If N 1 ad there exists a l 1 such that s0 l l, d l 1 ad d ( s0) ; the d N. (ii) If N 1 ad there exists a l 1 such that s0 l l, d l1ad d ( s0 ) ; the d N (iii) If N 1ad there exists a 1 l (iv) If N 1ad there exists a 1 l such that such that s s, d l1ad d ( s0 ) ; the d N. 0 l, d l1ad d ( s0 ), the d N. 0 l Proof: Tae oly case (i) as a example to show d ( s0 ) d ( N 1). It ca see that (l 1) ( (l l)) a(l 1) (l l) 1 ( a(l 1) (l l)) 1 3al a 3l 1l 1 (l 1)(a l 1) (l 1) ( (l l)) a(l 1) (l l) 1 ( a(l 1) (l l)) 1 3al a 3l 1l 1 (l 1)(a l 1) DOI: / Page

4 Seed ad Sieve of Odd Composite Numbers with Applicatios I Factorizatio of Itegers Now estimate the bouds of l for each case. (i) : (l 1) ( (l l)) (l 1) ( (l l)) 4l 14 l ( 1) 0 14 ( 7) 114( 1) 11 l 1 14 (ii) (l 1) ( (l l)) (l 1) ( (l l)) 4l 14 l ( 1) ( 1) 1 l 1 14 (iii) (l 1) ( l ) 3l 1l 1 l 4l 1 l ( 1) ( 1) l (iv) (l 1) ( l ) 3l 1l 1 l 4l 1 l ( 1) ( 1) l Characteristics of Odd Composite Numbers 1 Puttig cosecutive odd umbers oe after aother to form a sequece O by O={1,3, 5, 7, 9, 11,13,15, 17,, Oe ca see that, by taig 1 to be of the form +1, the umber before 3_m is of the form +1 ad the umber after 3_m is of the form -1. It also sees that, such a law is valid for arbitrary sub-sequece O s of O, say O s ={7,9,11,. For this reaso, O ca be partitioed by a series of group-uits by O {1,3,5,7,9,11,13,15,17,19,1,3,5,7,9,31,33,35,37,39,41,43,45,47,49,51,53, g1 g g3 g4 g5 g g7 g8 g9 55,57,59,1,3,5,7,9,71,73,75,77, 79,81,83,85,87,89, 91, 93, 95, 97, 99,101, g10 g11 g1 g13 g14 g15 g1 g17 103,105,107,109,111,113,115,117,119,11,13,15,17,19,131,133,135,137, g18 g19 g0 g 1 g g3 139,141, 143,145,147,149,151,153,155,157,159,11,13,15,17,19,171,173,... g4 g5 g g7 g8 g9 Deote g( i,1), g( i,) ad ( i,3) i 1,,... g to express the first, the secod ad the third umber of group g i ; the it is true for g( i,1) ( i 1) 1 g( i,) g( i1,3) g( i,) ( i 1) 3 g( i,1) g( i,3) g( i,3) ( i 1) 5 g( i,) g( i1,1) g( i, j) ( i 1) j 1, j 1,,3 i ( g( i,3) 1) / ( g( i,1) 5) / j 3 (1 g( i, j) )(mod3) The the followig Theorem 4 holds. Theorem 4: For the sequece O, the followig statemets are true. (i) For arbitrary m Z, m 0(mod5) leads to 5 ( m,1) ad 5( m 1,3). (ii) Arbitrary m, Z yield g g, j 1,,3 ( m, j) ( m, j) ad arbitrary iteger m>1 g g, g g, ( m,1) (m 10m 5,1) ( m,) (m m,) 3 g( m,3) g ad g 3 (m m1,1) ( m,3) g(3m 18m 3 m,3) (iii) Arbitrary m1 g (, ) g (, ) ( m ), where j 1,,3. (iv) For arbitrary, Z 0 m j, it holds j DOI: / Page (9)

5 Seed ad Sieve of Odd Composite Numbers with Applicatios I Factorizatio of Itegers g g g ( g )( g ) g ( m,1) (,1) (,1) ( m,1) (,1) (*,1) g( m,3) g(,3) g(,1) ( g( m,3) )( g(,3) ) g(*,1) where g (*,1) meas the first umber of certai group. (v) Every m 8 g( m,1) ( 1)( 7), Z ad geerally, every m a( 1) ( ) 1 g( m,1) ( 1)( a 1), a, Z (vi) Every m 4 1 g( m,1) ( 1)( 5), Z ad geerally, every a( 1) ( ) g( m,1) ( 1)( a 1), a, Z (vii) Every m 1 g( m,3) ( 1)( 5), Z ad geerally, every a( 1) g( m,3) ( 1)( a 1), a, Z (viii) For arbitrary itegers m,, Z ad m 1, g ( ) ( mj, ) m or ( m) ( mj, ) leads to g( m, j) (, j) or ( m) ( j, ) respectively, where j 1,,3 ; particularly, if m(mod ) ad ( mj, ), the (, j) ad thus ( m, g( m, j) ) (, j) ; cosequetly, g ( m,1) ad g(,1) are of the same form of either (+1)(l+1) or (-1)(l-1). (ix) For arbitrary itegers m,, Z ad m 1, if g( m, j) (, j), the g( m, j) ( g( m, j), j) ; if g( m, j ) (, j ), the g ( m, jm ) ( g( m, j ), j ) m (x) For arbitrary itegers m, Z ad m 1, it holds g(, j ) ( m, j ), where m ad j m are give by m r g(, j )(mod3) g(, j ) j m 3 3 (10) jm ( r j)(mod3) i which Z. Proof: (i), (ii) ad (iii) are from basic formulas (9), Lemma 3 ad direct calculatios. (iv) is from Lemma 3; (v),(vi) ad (vii) are from Theorem 3; (viii) is from (iii); (ix) is from (ii) ad (viii). (x) is from Lemma 4. Theorem 5: For arbitrary itegers m,, Z ad m 1, suppose g( m, j ) (, jm) ad g(, j ) g ( m, j m ) ; the g(, j ) (, j ) if g(, ) g(, ). j j Proof: The coditio that g(, ) g(, ) meas there are umbers betwee g(, j ) ad g (, ), thus j m j m cosider two eighborig umbers first, say g( i,1) ad g ( i,). By lemma 4, the ( g ( i,1) ) th umber that is couted from g( i,) is a multiple p of g ( i,1), ad the ( g ( i,) ) th umber that is couted from g( i,), amely the oe ext to g ( i,), is a multiple q of g ( i,). That is to say, q is the 3 rd umber followig p. Figure 1 ca ituitively depicts the case. Ad so forth, the first multiple s of g(, j ) is the (3) th umber followig the first multiple t of g (, ). Cosequetly, if g( m, j ) (, jm) ad g(, j ) g(, j ) the g (, j ) (, j ), as depicted by figure. m j m mj m Fig 1. Neighborig umbers ad their first multiples DOI: / Page

6 Seed ad Sieve of Odd Composite Numbers with Applicatios I Factorizatio of Itegers Fig. Mappig of two odd umbers to their first multiples IV. Experimets ad Examples Usig the theorems proved previously, approaches to factorize a big composite umber ca be derived out. First, Theorem 1, ad 3 provide a id of direct search approach such that fidig a x to satisfy (x 1) ( x) or (x 1) ( x) for big odd umber N 1, or fidig a x to satisfy (x 1) ( x) or (x 1) ( x) for big odd umber N 1. Either Theorem or 3 shows that the searchig processes is limited although they might be of very log time. Secodly, Theorem 4 provides aother ew searchig approach that fid a m such that ( 3 g( m,1) (m 10m 5)) or g ( (3m 18m 3 m)) ( m,3). I the ed, it ca see that Theorem 5 provides a possible iquiry approach to fid a divisor of the umber 1 by iquirig its eighborig composite umber. This approach is ideed a fresh oe ad is of very high efficiecy, which will be specially itroduced i my followig article. The followig examples are respectively about to show approaches metioed before. The examples 1 to 5 are based o Theorem ad 3; the example 5 ad are based o Theorem 4 ad the last oe is based o Theorem Example 1: Let N=993; the 993=499-1;so =499 ad l 1. It ca see that l 7 (l 1 41) ((499 7) 49) 1. Hece 41 must be a factor of 993 as the fact shows 993= Example : Let N=4573;the 4573=7+1;so =7 ad l 15. It ca see that l 3 (l 1 17) ((7 3) 75) 45. Hece 17 must be a factor of 4573 as the fact shows 4573= Example 3: Let N 1; the N=341+1;so =341 ad ((341 4) 345) Hece 3 must be a factor of 1 as the fact shows Example 4: Let N 7891 ; the = 1315 ((1315 ) 187) l 10. It ca see that 4 (l1 3) which says as the fact shows 7891=1307 Example 5: Let N 341; the = 1315 l 1 7 ;It ca see that l ((l1) 13) l 1 1 ;It ca see that l 5 ((l1) 9) (( ) 3770) 130 Example : Let N ; the N= ad it has factor of the form x 1. Note that N= which says as the fact shows 341= = So eed to fid a m i g(*,3) such that g m m m. 3 ( m,3) ( ( )) 3 Computatio shows that ( g(10,3) 59) ( ( ) ) 943 ; hece, Example 7: Let N ; the N= = ad N has factors of the form either x+1 or x-1. Need to fid a m i g(*,1) such that g( m,1) ( (m 10m 5)). Computatio shows that g ) 4485 ; hece, ( (349,1) 089) ( ( ) Example 8: Let N ; sice ( 3-1)-( 11-1)= 11 ( 1-1)= = ad 11-1=389, test 3+ ad the fid DOI: / Page

7 Seed ad Sieve of Odd Composite Numbers with Applicatios I Factorizatio of Itegers V. Coclusios Factorizatio of big itegers has bee a research topic i fields of iformatio security. Up to ow, huma beigs have developed may valuated approaches. Nevertheless, ew approaches are still ecessary for both scietific research ad techical demads. By defiig seeds of composite umber ad costructig sieve of odd composite umbers, this article turs factorizatio of a big odd composite umber to that of its seed, ad to fidig its eighborig composite umber. Experimets show that the ew approaches are valid ad practical. The author this that, by aids with other methods, the ew approaches are sure to be effective oes. Acowledgemets The research wor is supported by the atioal Miistry of sciece ad techology uder project 013GA78005, Departmet of Guagdog Sciece ad Techology uder projects 015A ad 015A , Fosha Bureau of Sciece ad Techology uder projects 01AG100311, Special Iovative Projects 014KTSCX15, 014SFKC30 ad 014QTLXXM4 from Guagdog Educatio Departmet. The authors sicerely preset thas to them all. Refereces [1] L E Dicso. History of the Theory of Numbers, Chelesea publishig Compay, New Yor.1971 [] S Sarai,D Gadear,U Gaiwad. A Overview to Iteger Factorizatio ad RSA i Cryptography, Iteratioal Joural for Advace Research i Egieerig ad Techology (IJARET),014,(IX):1-7 [3] XX Liu, XX Zou ad JL Ta. Survey of large iteger factorizatio algorithms, Applicatio Research of Computers,014,13(11): [4] WANG Xigbo. Valuated Biary Tree: A New Approach i Study of Itegers, Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR), Volume 4, Issue 3, March 01, PP 3-7 [5] CD Pa ad CB Pa, Elemetary Number Theory(3rd Editio), Press of Peig Uiversity, 013,pp- [] WANG Xigbo. New Costructive Approach to Solve Problems of Itegers' Divisibility, Asia Joural of Fuzzy ad Applied Mathematics, 014, (3):74-8 DOI: / Page