Introducing Sieve of Eratosthenes as a Theorem
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1 ISSN(Ole 9-8 ISSN (Prt - Iteratoal Joural of Iovatve Research Scece Egeerg ad echolog (A Hgh Imact Factor & UGC Aroved Joural Webste wwwrsetcom Vol Issue 9 Setember Itroducg Seve of Eratosthees as a heorem E Hassa Assstat Professor Deartmet of athematcs College of Scece ad Arts af Uverst Raa Saud Araba ABSRAC I ths aer we establsh a theorem usg same techque seve of Eratosthees to fd uow rme umbers a gve rage also the gve rage we rove there exst rme umbers dfferet from ow rme umbers KEYWORDS Seve of Eratosthees rme umbers bouded multles I INRODUCION Seve of Eratosthees s a ver smle acet algorthm for fdg all rme umbers the rage from to a gve umber Itall we have the set of all the umbers S At each ste we choose the smallest umber the set ad remove all ts multles It s suffcet to remove ol multles of rme umbers ot exceedg We start frst wth the smallest elemet the set we remove multles of from the set S whch are 8 At secod ste we choose smallest umber of the remag elemets of the set S We remove multles of from remag elemets of the set S whch are 9 At thrd ste we choose smallest umber of the remag elemets of the set S We remove multles of from remag elemets of the set Swhch are Cotue ths rocess at each ste we choose the smallest umber ad remove all ts multles It s suffcet to remove each ste ol multles of rmes umbers ot exceedg I ths wa all comoste umbers wll be removed from the set S ad rema all rme umbers ths set Seve of Eratosthees s a well ow algorthm for fdg rme umbers betwee ad several authors reseted dfferet algorthmc forms see eg [89] I ths aer we defe the cocet of bouded multles of a teger a set ad rove some roostos related to bouded multles we use these roostos to rove the ma result ths aer he ma result ca be used to fd rme umbers betwee ow two tegers Suose all rme umbers ow to us are ad s the largest rme umber ow to us How to fd rme umbers betwee ad where s a alcato of the ma result also we rove there exsts at least oe rme umber betwee ad I the ma result we use the same techque seve of Eratosthees algorthm here we remove ow rme umbers ad ther multles the remag umbers betwee ad are uow rme umbers ad ther roducts to a owers whch ca be wrtte as tegers II RELAED WORK Here we troduce the cocet of fte multle wth some examles ad roostos Defto Let m (subset of atural umbers wth m elemets ad greatest elemet m ad a ostve teger we called the set a ax ax x bo a set uded multles of teger a the set Corght to IJIRSE DOI8/IJIRSE9 8
2 ISSN(Ole 9-8 ISSN (Prt - Iteratoal Joural of Iovatve Research Scece Egeerg ad echolog (A Hgh Imact Factor & UGC Aroved Joural Webste wwwrsetcom Vol Issue 9 Setember Examle Let ad From Dfto for all set we have Remar Let m ad a ostve teger the set a aaa am from Defto a ad m m Remar B the set of rme umbers we mea ad so o Examle Let tegers tegers 98 tegers x x teger ot multle of teger ot multle of ( z z ( ot multleof ad ot multleof 9 ( 8 8 rme umber teger 8 a ax ax x Proosto Let a multles of teger a the set b s ostve teger ( b a s multles of teger b b the set a ( ba s multles of teger ba the set the ( b a ( ba ad ( ab ( ba Proof From Defto we have ( b a bax baxa ax a bax bax a ax a bax bax x ( ba ( ab abx abx x bax bax x ( ba Proosto Let a s bouded fte multles of teger a b elemets of the set ad b s bouded fte multles of teger b b elemets of the set f b s multle of a the b a ad f a a the Proof Suose b s multle of a the there exsts ostve teger c such that b ac From Dfto there exsts m m such that m a aaa m a b bb b m b ad aaa m a m bbb mb where m m m m Sce cacaca m ca bbb m b aaa ma ad c ca a we have m m the a cacaca m ca aaa m therefore b a ad Corght to IJIRSE DOI8/IJIRSE9 88
3 ISSN(Ole 9-8 ISSN (Prt - Iteratoal Joural of Iovatve Research Scece Egeerg ad echolog (A Hgh Imact Factor & UGC Aroved Joural Webste wwwrsetcom Vol Issue 9 Setember Suose a a from Defto a a a a m a such that m ad a aaa t a such that t where m m t t are ostve teger such that m t t m m m m t t t the we a a have t a t m m t m ad a a a t a t a t m a m m therefore t m ad a a a Proosto 8 Let a s bouded fte multles of teger a b elemets of the set ad ab ad s bouded fte multles of teger ab b elemets of the set where b s a ostve teger If ab a the we have b Proof Usg Proosto ad let a ab therefore f ab a the we have b Proosto 9 If a s bouded fte multles of teger a b elemets of the set ad for a ostve teger b whch satsfes b s bouded fte multles the we have ( ab a Proof From Proosto we have ( b a ( ba Usg Dftowe have ( ba ( b a bax bax a ax a a III AIN RESUL I ext theorem we wll rove there exsts at least oe rme umber betwee ad heorem For rme umbers let the there exsts at least oe rme umber q such that q Proof Gve the ca ot be dvded b a elemet of the set because resdue of dvso b equal ad for all Sce a teger greater tha oe ether rme or comoste of fte umber of rmes ad there are ftel ma rmes for s rme umber the we have therefore the theorem holds f s comoste the where are rme umbers such that ad are tegers ot all equal zero (f all of are equal zero the If the there exsts rme umber such that If the there exsts rme umber such that Geerall f the there exsts rme umber such that for all therefore the theorem holds for comoste herefore exsts at least oe rme umber q such that q Examle Let ad the there exst such that Examle Let ad the there exst such that Now we are read to rove our ma result heorem Let be rme umbers ad ad for rme umber let t t t (bouded multles of the teger b elemets of the set the for Corght to IJIRSE DOI8/IJIRSE9 89
4 ISSN(Ole 9-8 ISSN (Prt - Iteratoal Joural of Iovatve Research Scece Egeerg ad echolog (A Hgh Imact Factor & UGC Aroved Joural Webste wwwrsetcom Vol Issue 9 Setember Corght to IJIRSE DOI8/IJIRSE9 8 rme umbers such that the we have tegers Proof Sce ad b usg heorem there exsts at least oe rme umber q such that q suose that all rme umbers betwee ad are such that For all x we ca wrte x tegers For all we ca sa ad ot multle of we ca wrte tegers For all ( we ca sa ad ot multle of or multle of we ca wrte tegers Cotue ths rocess we ca sa ad ot multle of a elemet of the set we ca wrte tegers herefore tegers Examle Let ad (( ad 9 We have ( ad 9 ( Examle Let ad ad 9 We have ( 9 ( ( teger IV CONCLUSION Suose all rme umbers ow to us are ad s the largest rme umber ow to us Let ad f we al heorem we ca dscover a set of rme umbers sce tegers the two smallest elemets the
5 ISSN(Ole 9-8 ISSN (Prt - Iteratoal Joural of Iovatve Research Scece Egeerg ad echolog (A Hgh Imact Factor & UGC Aroved Joural Webste wwwrsetcom Vol Issue 9 Setember set are herefore s smallest elemet of dscovered set of rme umbers For the set tegers the two smallest elemets the set are herefore the two smallest elemets of dscovered set of rme umbers are ad such that Also tegers the two smallest elemets the set are herefore the smallest three elemets of dscovered set of rme umbers are such that Cotue ths rocess the dscovered set of rmes umbers wll be such that REFERENCES [] Sha D "Class umber a theor of factorzato ad geera Proc Sm I Pure athematcs 99 Amer ath Soc Provdece RI (9 - [] arso HG "Some ew uer bouds o the geerato of rme umbers" Comm AC 9-9(9 [] Carter Ba ad Rchard Hudso "he segmeted seve of Eratosthees ad rmes arthmetc rogressos to " BI - (9 [] Gres D ad sra J "A lear seve algorthm for fdg rme umbers" Commucatos of the AC ( 999- (98 [] Bohar S H "ult rocessg the seve of Eratosthees" IEEE Comut ( -8 (98 [] Prtchard P "Exlag the wheel seve" Acta Iformatca ( -8 (98 [] Prtchard P "Fast comact rme umbersseves" J Algorthms ( - (98 [8] Prtchard P "Lear rme umber seves A faml tree" Scece of comuter rogrammg (9 - (98 [9] Soreso J ad Parberr I"wo fast arallel rme umber seves" Iformato ad Comutato -(99 [] Le Veque WJ ocs Number heore Volume I Addso-Wesle Readg ass 9 [] Huter J Number heor (Olver ad Bod 9 [] Irelad K ad Rose A Classcal Itroducto to oder Number heor Secod ed Srger Verlag New Yor 99 Corght to IJIRSE DOI8/IJIRSE9 8
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