ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS

Transcription

1 Joual of Pue ad Alied Mathematics: Advaces ad Alicatios Volume 0 Numbe 03 Pages 5-58 ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS ALI H HAKAMI Deatmet of Mathematics Faculty of Sciece Jaza Uivesity P O Bo 77 Jaza 5 Saudi Aabia aalhaami@azauedusa Abstact Euclid s theoem is a fudametal statemet i umbe theoy that assets that thee ae ifiitely may ime umbes I this ae we shall give a ew aalytic techique to ove this well ow theoem Also we shall use Euclid s oof to show some facts Itoductio Thee ae seveal oofs fo the followig well ow theoem: Theoem The umbe of imes is ifiite That is thee is o ed to the sequece of imes Mathematics Subect Classificatio: Pimay 0A05 A A5; Secoday A07 A05 A5 Keywods ad hases: ime ifiite of ime (IP) Euclid s theoem Euclid s oof Eule s oof of IP Received Jue Scietific Advaces Publishes

2 5 ALI H HAKAMI Theoem fist aeas i the wos of Euclid [] which has the advatage of deedig o little beyod the defiitio of the imes Aothe oof fo Theoem by the Swiss mathematicia Leohad Eule [7] elies o the fudametal theoem of aithmetic see fo eamle [8] that evey itege has a uique ime factoizatio We shall give these two oofs i the et sectio Hillel [5] itoduced a oof fo Theoem by usig oit-set toology ad Piasco [3] by usig iclusio-eclusio icile the umbe of ositive iteges less tha o equal to that ae divisible by oe of the smallest N imes N Whag [] has ecetly ublished his oof fo Theoem by cotadictio with the de Poligac s fomula! f ( ) ime whee is ay ositive itege ad f ( ) Theoem ca be also oved by usig Eule s totiet fuctio ϕ (see fo eamle [6]) I this oof oe eed the followig oety: Fo 3 ϕ( ) is a eve itege Lastly oe ca oof Theoem by usig the iatioality of He eed to amly the followig fomula which was also discoveed by Eule imes: Euclid s ad Eule s Poof That the Numbe of Pimes is Ifiite Euclid s oof of Theoem We assume thee ae oly the imes ad o moe Coside the umbe Q Now is eithe ime o else has a ime facto If Q is ime we have a cotadictio sice ae suosed to be all imes But ay ime facto of Q must be diffeet fom all of sice it is easy to see that oe of the s ca divided Q This agai cotadictios the assumtio that comise all the imes

3 ON EUCLID S AND EULER S PROOF THAT 53 It is ofte eoeously eoted the Euclid oved this esult by cotadictio begiig with the assumtio that the set iitially cosideed cotais all ime umbes o that it cotais ecisely the smallest imes athe tha ay abitay fiite set of imes (see []) Although the oof as a whole is ot by cotadictio i that it does ot begi by assumig that oly fiitely may imes eist thee is a ove by cotadictio withi it that is the oof that oe of the iitially cosideed imes ca divide the umbe called Q above Eule s oof of Theoem Eule gave a moe sohisticate oof of Theoem Agai we assume ae all the imes The if is ay ositive itege we may choose big eough so that all the tems 3 aea whe we multily out the oduct I fact iceasig meely adds i moe tems Fo eamle we get by choosig fom the fist facto 3 fom the secod ad fom all the othes (assumig that ad 3 ) Now by the fomula fo the sum of geometic ogessio ( ) < We see that fo ay the sum 3 caot eceed But this cotadicts the divegece of the hamoic seies 3 So thee is a ifiite umbe of imes

4 ALI H HAKAMI 5 3 Theoem ad the Hamoic Seies Ou techique to ove Theoem is to ove that the hamoic seies is diveges To this ed we eed the followig lemma: Lemma (i) Fo ay ositive itege (ii) Let the i-th ime be i ad let N be the set of all ositive iteges all of whose ime factos ae The N N (iii) The limit as (iv) If 0 C (C is costat) the C e C Poof (i) Let ( ) S be eeseted by Clealy ( ) ( ) 3 S S If we tae a limit i the left had side ad the ight had side we will get the followig equality used i the followig equality:

5 ON EUCLID S AND EULER S PROOF THAT 55 (ii) We obseve that the fudametal theoem of aithmetic (see [8]) imlies N i i i ad N i i i Now (i) gives i i i i i Multilyig these idetities fo i fom to we get the statemet of (ii) (iii) The sequece s i i is iceasig If it is bouded the so is N N (sice is bouded by N ad the seies is coveget) But this is ot the case sice fo sufficietly lage ay N is bigge tha ay atial sum of seies is diveget Thus s is ot bouded that is N lim s ad the hamoic (iv) Coside a fuctio f ( ) e We have i f ( ) e 0 fo 0 Thus f ( ) is iceasig fo 0 Sice f ( 0 ) 0 we get that f ( ) 0 fo ay ositive

6 56 ALI H HAKAMI Ou oof of Theoem Use Lemma (iii) ad (iv) we have e e e e Thus the sequece t gows without boud This meas the umbe of imes is ifiite Some Alicatio I this sectio we shall adat the oof of Euclid to show the followig facts: () The -th ime does ot eceed fo ay ositive itege () Fo ay ositive itege ( ) log log (hee ( ) deotes fo the umbe of imes ) (3) Thee ae ifiitely may imes of the fom 3 N Fo () we eed to use Euclid s oof ad geealized iductio so 0 whe the fist ime is Suose that fo ay ad ay We have to show that Reasoig as i Euclid s oof thee is a ime which divides ( ) ie ( ) ad the the ( ) th ime which is smalle o equal tha the ime is such that P usig the iductio hyothesis Fo ay itege m ad ay eal umbe we have that

7 ON EUCLID S AND EULER S PROOF THAT 57 0 m m m (which follows easily by multilyig by ( ) o both sides) The 0 Ad elacig i the equatio above we have ( ) fo ay Fo () usig () we have that as the -th ime is at most ae Let be the uique itege such that < The ( ) ad usig the above coditio we have that log log log which imlies that ( ) log log Fially fo (3) we suose that thee ae oly fiitely may imes of the fom 3 say We coside the itege 3 N

8 58 ALI H HAKAMI We have to show that thee is a ime of the fom 3 dividig N Suose that all imes dividig N ae of the fom 3 ie N is a oduct of imes of the fom 3 The as the set of those imes is closed ude oduct N would also be of the fom 3 which is i false as N 3( ) 3 3( ) i Thee is the a ime of the fom 3 dividig N If i fo some i i the divides ( ) N which is imossible This is a i cotadictio ad it is shows that thee ae ifiitely may imes of the fom 3 Refeeces [] J Williamso (taslato ad commetato) The Elemets of Euclid With Dissetatios Claedo Pess Ofod 78 age 63 [] Michael Hady ad Catheie Woodgold Pime simlicity Mathematical Itelligece 3() (009) -5 [3] J P Piasco New oofs of Euclid s ad Eule s theoems Ameica Mathematical Mothly 6() (009) 7-73 [] J P Whag Aothe oof of the ifiitude of the ime umbes Ameica Mathematical Mothly 7() (00) 8 [5] Hillel O the ifiitude of imes Ameica Mathematical Mothly 6(5) (955) 353 [6] Divid M Buto Elemetay Numbe Theoy Sith Editio McGaw-Hill 007 [7] G H Hady ad E M Wight A Itoductio to the Theoy of Numbes Ofod Uivesity Pess Ofod 980 [8] I Nive H S Zucema ad H L Motogomey A Itoductio to the Theoy of Numbes Wiley New Yo 99 g