Sebastián Martín Ruiz. Applications of Smarandache Function, and Prime and Coprime Functions

Transcription

1 Sebastá Martí Ruz Alcatos of Saradache Fucto ad Pre ad Core Fuctos 0 C L f L otherwse are core ubers Aerca Research Press Rehoboth 00

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3 Sebastá Martí Ruz Avda. De Regla 43 Choa 550 Cadz Sa Alcatos of Saradache Fucto ad Pre ad Core Fuctos Aerca Research Press Rehoboth 00

4 Ths boo ca be ordered crofl forat fro: Bell ad Howell Co. Uversty of Mcrofl Iteratoal 300 N. Zeeb Road P.O. Box 346 A Arbor MI USA Tel.: Custoer Servce htt://wwwlb.u.co/bod/search/basc Boos o Dead Coyrght 00 by Aerca Research Press Rehoboth Box 4 NM 873 USA htt:// ISBN: Stadard Address Nuber Prted the Uted States of Aerca 3

5 Cotets: Chater : Saradache Fucto aled to erfect ubers Chater : A result obtaed usg the Saradache Fucto Chater 3: A Cogruece wth the Saradache Fucto Chater 4: A fuctoal recurrece to obta the re ubers usg the Saradache re fucto Chater 5: The geeral ter of the re uber sequece ad the Saradache re fucto Chater 6:xressos of the Saradache Core Fucto Chater 7: New Pre Nubers 4

6 Chater : Saradache fucto aled to erfect ubers The Saradache fucto s defed as follows: S the sallest ostve teger such that S! s dvsble by. [] I ths artcle we are gog to see that the value ths fucto taes whe s a erfect uber of the for beg a re uber. Lea : Let such that: whe s a odd re uber ad a teger 0 e! 3 L log where e! s the exoet of the re uber decoosto of!. x s the greatest teger less tha or equal to x. Oe has that S. Deostrato: Gve that GCD GCD greatest coo dvsor oe has that S ax{ S S } S. Therefore S. If we rove that! s dvsble by the oe would have the equalty. e!! where s the clear that s! for whch: e! L e s! s th re of the re uber decoosto of! e s. It s e! e! e s! s! L 5

7 Fro where oe ca deduce that:! e! e! e! s L s s a ostve teger sce e! 0. Therefore oe has that S Proosto: If s a erfect uber of the for wth s a ostve teger re oe has that S. Deostrato: For the Lea t s suffcet to rove that e!. If we ca rove that: we wll have roof of the roosto sce: As s a teger oe has that e! Provg s the sae as rovg at the sae te sce s teger s equvalet to rovg. I order to rove we ay cosder the fucto: uber. x f x x x real Ths fucto ay be derved ad ts dervate s f x x l. f wll be creasg whe x l > 0 resolvg x: ll x > '587 l I artcular f wll be creasg x. Therefore x f x f 0 that s to say x 0 x x. 6

8 Therefore: teger. Ad thus s roved the roosto. XAMPLS: 6 3 S S S S887 Refereces: [] C. Dutrescu ad R. Müller: To joy s a Peraet Cooet of Matheatcs. SMARANDACH NOTIONS JOURNAL Vol. 9 No

9 8 Chater : A result obtaed usg the Saradache Fucto Saradache Fucto s defed as followed: SThe sallest ostve teger so that S! s dvsble by. [] Let s see the value whch such fucto taes for wth teger ad re uber. To do so a Lea requred. Lea Ν log L where x gves the greatest teger less tha or equal to x. Proof: Let s see the frst lace the value tae by log. If : < ad therefore log log <. Ad f : [ ] log log log As a result: [ ] log < therefore: [ ] log f Now let s see the value whch t taes for :

10 If : If < : Let s see what s the value of the su: Therefore: Proosto: re uber : S Proof: Havg e exoet of the re uber the re decoosto of. We get: e L 3 log 9

11 0 Ad usg the lea we have [ ] e log! L Therefore: Ν Ν ad!! Ad : S Refereces: [] C. Dutrescu ad R. Müller: To joy s a Peraet Cooet of Matheatcs. SMARANDACH NOTIONS JOURNAL VOL 9: No

12 Chater 3: A Cogruece wth the Saradache fucto Saradache s fucto s defed thus: S s the sallest teger such that S! s dvsble by. [] I ths artcle we are gog to loo at the value that has S od For all teger 97. S - S - od

13 S - S - od

14 S - S - od Oe ca see fro the table that there are oly 4 excetos for 97 3

15 We ca see detal the 4 excetos a table: 8 7 S 8 -h5 od S 5 -h7 od S 68 -h35 od S 9 -h47 od 9 Oe ca observe these 4 cases that wth s a re ad ore over S od UNSOLVD QUSTION: Oe ca obta a geeral forula that gves us fucto of the value S od for all ostve teger values of?. Referece: [] Saradache Notos Joural Vol. 9 No

16 Chater 4: A fuctoal recurrece to obta the re ubers usg the Saradache re fucto. Theore: We are cosderg the fucto: For teger: F j j j oe has: F for all where { } are the re ubers ad x s the greatest teger less tha or equal to x. Observe that the owledge of oly deeds o owledge of ad the owledge of the fore res s uecessary. Proof: Suose that we have foud a fucto P wth the followg roerty: f s cooste P 0 f s re Ths fucto s called Saradache re fucto.ref. Cosder the followg roduct: P If < < P sce : are all coostes. 5

17 If P 0 sce P 0 Here s the su: P P P The secod su s zero sce all roducts have the factor P 0. Therefore we have the followg recurrece relato: P Let s ow see we ca fd P wth the ased roerty. Cosder: j j 0 s s j j ot j L We deduce of ths relato: d j j j where d s the uber of dvsors of. 6

18 7 If s re d therefore: 0 d If s cooste > d therefore: 0 < < d d Therefore we have obtaed the Saradache Pre Fucto P whch s: j j P j teger Wth ths the theore s already roved. Refereces: []. Burto Saradache Pre ad Core fuctos. []F. Saradache Collected Paers Vol II Kshev Uversty Press Kshev 997.

19 Chater 5: The geeral ter of the re uber sequece ad the Saradache re fucto. Let s cosder the fucto d uber of dvsors of the ostve teger uber. We have foud the followg exresso for ths fucto: x Floor[x] d We roved ths exresso the artcle A fuctoal recurrece to obta the re ubers usg the Saradache Pre Fucto. We deduce that the folowg fucto: d G Ths fucto s called the Saradache Pre Fucto Referece It taes the ext values: 0 G f f s s re cooste Let s cosder ow π uber of re ubers saller or equal tha. It s sle to rove that: π G 8

20 Let s have too: If If C π 0 π We wll see what codtos have to carry C. Therefore we have the followg exresso for -th re uber: C π If we obta C that oly deeds o ths exresso wll be the geeral ter of the re ubers sequece sce π s fucto wth G ad G does wth d that s exressed fucto wth too. Therefore the exresso oly deeds o. Let s cosder C log Sce log fro of a certa 0 t wll be true that log If 0 t s ot too bg we ca rove that the equalty s true for saller or equal values tha 0. It s ecessary to that: log π If we chec the equalty: π log < 9

21 0 We wll obta that: ; < C C C C π π π We ca exeretaly chec ths last equalty sayg that t checs for a lot of values ad the dfferece teds to crease wch aes to th that t s true for all. Therefore f we rove that the ad equaltes are true for all whch sees to be very robable; we wll have that the geeral ter of the re ubers sequece s: log / / j j s j s j s j Referece: []. Burto Saradache Pre ad Core Fuctos Htt:// [] F. Saradache Collected Paers Vol. II Kshev Uversty Press.

22 Chater 6: xressos of the Saradache Core Fucto Saradache Core fucto s defed ths way: otherwse ubers core are f C 0 L L We see two exressos of the Saradache Core Fucto for. XPRSSION : lc C x the bggest teger uber saller or equal tha x. If are core ubers: lc therefore: 0 0 C If are t core ubers: 0 < < < C lc lc XPRSSION : > > ' ' ' ' d d d d d d d d d d C

23 If are core ubers the d d' d d' d d' d d ' d > d ' > 0 < < C d d' d d 0 If are t core ubers d d' d > d' > C XPRSSION 3: Saradache Core Fucto for : C L GCD L If L are core ubers: GCD L C L 0 If L are t core ubers: GCD L > 0 < < C L GCD GCD Refereces:.. Burto Saradache Pre ad Core Fucto. F. Saradache Collected Paers Vol II.. 37Kshev Uversty Press.

24 Chater 7: New Pre Nubers I have foud soe ew re ubers usg the PROTH rogra of Yves Gallot. Ths rogra based o the followg theore: Proth Theore 878: Let N where <. If there s a teger uber a so that a N od N therefore N s re. The Proth roga s a test for ralty of greater ubers defed as b or b. The rogra s ade to loo for ubers of less tha dgts ad t s otzed for ubers of ore tha 000 dgts.. Usg ths Progra I have foud the followg re ubers: wth 370 dgts a 3 a wth 37 dgts a 5 a wth 37 dgts a 3 a 9363 wth 373 dgts a 5 a 7 Sce the exoets of the frst three ubers are Saradache uber S5345 we ca call ths tye of re ubers re ubers of Saradache. Heled by the MATHMATICA roga I have also foud ew re ubers whch are a varat of re ubers of Ferat. They are the followg: 3 3 for It s ortat to eto that for 7 the uber whch s obtaed has 00 dgts. 3

25 Chrs Nash has verfed the values 8 to 0 ths last oe beg a uber of dgts obtag that they are all cooste. All of the have a ty factor excet 3. Refereces:. Mcha Fleure Saradache Factors ad Reverse Factors Saradache Notos Joural Vol. 0 Chrs Caldwell The Pre Pages 4

26 A boo for eole who love ubers: Saradache Fucto aled to erfect ubers cogrueces. Also the Saradache Pre ad Core fuctos coecto wth the exressos of the re ubers. $5.95 5