γραμμική αλγεβρα

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Carl Friedrich Gauss (1777-1855)
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  1. 1. Gnwrzete ti grfw me arg rujm. Aut ofeletai kurwc sto ti den emai pot ikanopoihmnoc wc tou qw pei so to dunatn peris- stera me lgec lxeic, kai h sunoptik graf apaite pol perisstero qrno ap thn ekten graf. Carl Friedrich Gauss (1777-1855)
  2. 2. Mia Eisagwg sthn 'Algebra Deterh 'Ekdosh Dhmtrioc A. Brsoc Dhmtrioc I. Derizithc Iwnnhc P. Emmanoul Miqal P. Malikac Olumpa Talllh
  3. 3. [RO,LE]2 [LO] [RE] [CE,CO]
  4. 4. Prlogoc Aut to biblo apeujnetai kurwc se proptuqiakoc foithtc twn Majhmati- kn. Didskontac ep seir etn Majhmatik, brejkame apnanti se na shma- ntik paidagwgik prblhma: Ta Majhmatik -kai eidik h 'Algebra- enai ap th fsh touc afairetik. Eke gkeitai h dunamik touc all kai h sqedn kajolik qrhsimtht touc. Pc loipn epitugqnetai se na eisagwgik m- jhma 'Algebrac mia isorropa tou afairetiko stoiqeou me to sugkekrimno? Oi prospjeic mac gia to skop aut apotupnontai sto parn biblo. Krio qarakthristik thc apotpwshc autc enai ti katabljhke idiaterh prospjeia tsi ste afenc h afaresh na pragmatopoietai stadiak, dhlad na parqontai knhtra pou na dikaiologon thn eisagwg nwn ennoin, kai afetrou na epexhgontai me so to dunatn apl trpo oi shmantikc teqnikc pou qrhsimopoiontai stic apodexeic twn apotelesmtwn. Gia to skop aut, sto parn biblo uprqei nac megloc arijmc epexergasmnwn paradeigmtwn kai efarmogn. Epeid de pepojhs mac enai ti h melth twn Majhmatikn den armzei se jeatc all apaite energhtik summetoq, epiqeirsame na parembloume msa sto kemeno aplc erwtseic, ste na epibebainetai ap ton anagnsth kat pso qei katanosei tic prohghjesec twn erwtsewn nnoiec. Epshc, qoun sumperilhfje arketc askseic tsi ste o foithtc pou melet to biblo na enjarrnetai na qrhsimopoie qart kai molbi. H lxh 'Algebra enai mroc tou ttlou enc Arabiko keimnou tou 800 m.Q. to opoo anafertan stouc kannec gia th lsh twn exissewn kai mqri perpou ta msa tou 19ou aina h 'Algebra tan o kldoc twn Majhmatikn pou asqoleto me th melth twn exissewn. Den uprqei amfibola ti o roc dom katqei mia ap tic kuriterec jseic sta sgqrona Majhmatik. Prgmati, h swst katanhsh kai melth twn domn jewretai jemelidec ergaleo gia thn projhsh miac enopoihmnhc anptuxhc twn Majhmatikn. 'Ena ap ta shmantiktera edh domn apotele na snolo efodiasmno me ma perissterec prxeic. Smera, h 'Algebra 3
  5. 5. 4 Prlogoc enai h melth akribc autn twn domn. Gia to lgo aut, mpore kanec na pei dikaiologhmna ti h 'Algebra den apotele mno na mroc twn Majhmatikn, all ti pazei gia ta Majhmatik to rlo pou aut paixan kai pazoun sthn anptuxh twn jetikn episthmn. Enai koin pepojhsh ti h na antlhyh gia to rlo thc 'Algebrac pou epikrthse sth dekaeta tou 1920, h opoa sundetai me ta onmata thc Emmy Noether, twn Emil Artin, B.L. Van der Waerden, Hermann Weyl kai llwn epifann majhmatikn, bassthke sth melth jemeliwdn domn pwc oi O- mdec, oi Daktlioi, ta Prtupa kai ta Smata. Aut h antlhyh thc Sg- qronhc 'Algebrac (onomasa pou uiojetjhke gia aut th na tsh) odghse se mia jemelidh, kainotmo kai exairetik apodotik anptuxh thc 'Algebrac. Krio qarakthristik autc thc anptuxhc enai ti prin lga qrnia ljhkan shmantikc eikasec twn prohgomenwn ainwn, pou anafrontai se klasik Majhmatik. Wc na ttoio pardeigma prpei na anafroume th mnhmeidh eikasa tou Fermat, smfwna me thn opoa h exswsh xn + yn = zn den - qei kami akraia lsh (x, y, z), me ta x, y, z la difora tou mhdenc, tan to n 3. H eikasa aut apodeqthke ti isqei, ap ton Wiles, peita ap 350 qrnia. 'Alla qarakthristik paradegmata enai to 10o, 14o kai 17o pr- blhma tou Hilbert, pwc epshc kai oi eikasec tou Andre Weil. 'Ena de ap ta shmantiktera apotelsmata pou qei epiteuqje mqri smera sthn 'Alge- bra jewretai h apdeixh thc eikasac twn Burnside kai Galois, pou anafretai sthn dia th Jewra Omdwn, gia thn parxh thc taxinmhshc twn apln pe- perasmnwn omdwn. Apodeqjhke to 1980 kai apasqlhse touc megalterouc algebristc tou perasmnou aina, kurwc thc peridou 1950-1980. To biblo aut enai so to dunatn autodnamo kai kalptei th basik jewra twn daktulwn kai twn omdwn. H dom twn daktulwn exetzetai stic Enthtec 2 kai 3, en stic Enthtec 4 kai 5 anaptssetai h jewra twn omdwn. Oi Enthtec 2 kai 4 enai tsi domhmnec, otwc ste na enai anexrthtec h ma thc llhc. Sunepc, o foithtc enai elejeroc na meletsei touc daktu- louc prin met tic omdec. Sthn Enthta 1 meletntai oi basikc idithtec twn akerawn, kajc kai h nnoia thc monadikc paragontopohshc se ginmeno prtwn arijmn. 'Enac ap touc stqouc thc Enthtac 3 enai na katadexei thn nnoia aut geniktera stouc daktulouc monadikc paragontopohshc kai ka- tpin na thn efarmsei gia thn eplush sugkekrimnwn Diofantikn exissewn. Sthn Enthta 1 melettai epshc h arijmhtik twn isotimin, na jemelidec pardeigma pou anaptssetai geniktera stic Enthtec 2 kai 4. Sthn Enthta 4 melettai h nnoia thc summetrac, pou odhge fusiologik sthn nnoia thc omdac. Sth sunqeia, anaptssetai h basik jewra twn omdwn kai h na aut gnsh efarmzetai gia na lhfjon apotelsmata sth stoiqeidh jew-
  6. 6. 5 ra arijmn. Tloc, sthn Enthta 5 anaptssetai h nnoia thc drshc miac omdac, h opoa ousiastik genikeei thn nnoia thc summetrac. Wc basik efarmog autc thc nnoiac, apodeiknoume ta jewrmata tou Sylow, pou enai ap ta pio shmantik apotelsmata thc jewrac twn peperasmnwn omdwn. Epeid pisteoume ti na eisagwgik biblo ofelei na parqei kateujnseic ston endiafermeno anagnsth gia mia peraitrw embjunsh, uprqoun sto kemeno arketc parapompc se jmata thc Jewrac Arijmn, thc Metajetikc 'Algebrac, thc Jewrac Omdwn, k.. Ekfrzoume tic jermc euqaristec mac stic kurec Rza Gardrh kai Pph Mpolith gia thn taqea kai apotelesmatik daktulogrfhsh tou keimnou sto ssthma LaTeX. Epshc, euqaristome to foitht Baslh Pasqlh tou Tmmatc mac pou sunbale sth diamrfwsh twn sqhmtwn. Oi kriec allagc sth deterh aut kdosh, pra ap th dirjwsh lwn twn paroramtwn ta opoa uppesan sthn antlhy mac, enai oi exc: H Enthta 2 thc prthc kdoshc diaspsthke se do Enthtec, kajc oi tsseric teleuta- ec Pargrafo thc apotelon tra thn Enthta 3. Epiplon, llaxe h seir twn Paragrfwn 2.7 kai 2.8 kai prostjhke h Pargrafoc 2.9 (Epektseic Swmtwn kai Gewmetrikc Kataskeuc). Anamorfjhke h anptuxh thc lhc thc Enthtac 3 thc prthc kdoshc (pou apotele tra thn Enthta 4), me apotlesma na sugqwneujon oi Pargrafoi 3.3 kai 3.4 thc prthc kdoshc (sthn Pargrafo 4.3), oi Pargrafoi 3.5 kai 3.8 thc prthc kdoshc (sthn Pargrafo 4.5) kai na diaspaste h Pargrafoc 3.9 thc prthc kdoshc (stic Paragrfouc 4.7 kai 4.8). Oi suggrafec Ajna, Septmbrioc 2005
  7. 7. Perieqmena 7
  8. 8. 8 PERIEQOMENA
  9. 9. 1 Akraioi Sthn Enthta aut ja meletsoume idithtec tou sunlou twn akerawn arij- mn, Z. O skopc mac enai dittc. Afenc ja meletsoume tic idithtec tou Z pou ja qrhsimopoisoume se epmenec Enthtec kai afetrou ja eisgoume msw tou Z merikc jemelideic nnoiec thc 'Algebrac. 'Etsi, dnetai h eukaira ston anagnsth na meletsei arketc ap tic nnoiec thc 'Algebrac, pou emfa- nzontai paraktw, sthn eidik perptwsh twn akerawn. Se kami perptwsh h Enthta aut den apotele eisagwg ston plosio kldo thc Jewrac Arijmn. Ja jewrsoume gnwstc tic plon stoiqeideic idithtec thc prsjeshc kai tou pollaplasiasmo akerawn pwc kai thc ditaxhc, ... < 1 < 0 < 1 < ..., pou uprqei sto Z me tic opoec emaste exoikeiwmnoi ap ta prta majhti- k mac qrnia. Oi teqnikc twn apodexewn pou ja dome ed enai tupikc gia thn 'Algebra, prgma pou ja diapistsoume paraktw tan meletsoume daktulouc, smata kai omdec. Sto biblo aut ja qrhsimopoiome touc paraktw sumbolismoc. N = {0, 1, 2, . . . } To snolo twn fusikn arijmn. Z = {. . . , 1, 0, 1, . . . } To snolo twn akerawn arijmn. Q To snolo twn rhtn arijmn. R To snolo twn pragmatikn arijmn. C To snolo twn migadikn arijmn. Z>0 To snolo twn jetikn akerawn. Q>0 To snolo twn jetikn rhtn arijmn. R>0 To snolo twn jetikn pragmatikn arijmn. 9
  10. 10. 10 1 Akraioi 1.1 Majhmatik Epagwg, Diwnumiko Suntelestc Sthn Pargrafo aut ja upenjumsoume th mjodo apdeixhc pou ono- mazetai Majhmatik Epagwg. Epshc ja asqolhjome me touc diwnumikoc suntelestc. H afethra mac enai to epmeno axwma. 1.1.1 Axwma (Axwma Elaqstou). Kje mh ken snolo fusikn arij- mn periqei elqisto stoiqeo. Shmeinoume ti antstoiqh idithta den isqei stouc pragmatikoc rh- toc arijmoc. Gia pardeigma, to snolo {1, 1/2, 1/3, 1/4, ...} den periqei elqisto stoiqeo an kai enai uposnolo twn jetikn rhtn arijmn. Ap th ditaxh pou uprqei sto Z, parathrome ti kje mh ken snolo fusikn arijmn periqei monadik elqisto stoiqeo. 1.1.2 Jerhma (Majhmatik Epagwg). 'Estw ti gia kje fusik arijm n dnetai mia prtash P(n), pou afor ton n, ttoia ste 1) h P(0) alhjeei, kai 2) gia kje n, an h P(n) alhjeei, tte h P(n + 1) alhjeei. Tte h P(n) alhjeei gia kje fusik arijm n. Apdeixh. 'Estw A to uposnolo tou N pou apoteletai ap touc n gia touc opoouc h prtash P(n) den alhjeei, A = {n N|P(n) den alhjeei}. Ja dexoume ti to A enai ken. Ac upojsoume ti A = . Tte ap to Axwma Elaqstou to A periqei elqisto stoiqeo, stw m. Ap thn upjesh 1) qoume m > 0. Ap ton orism tou m petai ti h prtash P(m 1) alhjeei. Tte mwc h upjesh 2) dnei ti h P(m) alhjeei. Aut enai topo. Edame ti to prohgomeno Jerhma petai ap to Axwma Elaqstou. Mpore na apodeiqje ti to Jerhma thc Majhmatikc Epagwgc enai iso- dnamo me to Axwma Elaqstou ('Askhsh 8). H upjesh 1) sto prohgomeno Jerhma sunjwc onomzetai arqik bma thc epagwgc en h upjesh 2) epagwgik bma. Prin proqwrsoume se paradegmata, anafroume mia pa- rallag pou diafrei sto arqik bma thc epagwgc. 1.1.3 Jerhma (Majhmatik Epagwg me arqik bma ap to m). 'Estw m N. 'Estw ti gia kje fusik arijm n me n m dnetai mia prtash P(n), pou afor ton n, ttoia ste
  11. 11. 1.1. Majhmatik Epagwg, Diwnumiko Suntelestc 11 1) h P(m) alhjeei, kai 2) gia kje n m, an h P(n) alhjeei, tte h P(n + 1) alhjeei. Tte h P(n) alhjeei gia kje n N me n m. Apdeixh. H apdeixh enai parmoia me thn prohgomenh kai afnetai san skhsh. 1.1.4 Paradegmata. 1) 1 + 2 + + n = 1 2 n(n + 1) gia kje jetik akraio n. Qrhsimopoiome epagwg. Arqik bma: Gia n = 1, h apodeikta sqsh enai 1 = 1 2 1(1 + 1), pou isqei. Epagwgik bma: 'Estw ti isqei 1 + 2 + + n = 1 2 n(n + 1). P(n) Ja apodexoume ti isqei 1 + 2 + + (n + 1) = 1 2 (n + 1)(n + 2). P(n + 1) To arister mloc thc P(n + 1) grfetai me th bojeia thc P(n) 1 + 2 + + (n + 1) = 1 + 2 + + n + (n + 1) = 1 2 n(n + 1) + (n + 1). To dexi mloc thc parapnw isthtac enai 1 2 (n + 1)(n + 2), pou enai to zhtomeno. 2) 2n > n2 gia kje akraio n 5. Qrhsimopoiome epagwg. Gia n = 5 h apodeikta sqsh enai h 25 > 52, pou isqei. Upojtoume tra ti isqei 2n > n2 , P(n) pou n 5, kai ja apodexoume ti 2n+1 > (n + 1)2 . P(n + 1) Ap thn P(n) qoume ti 2n+1 = 2 2n > 2n2. Epiplon isqei 2n2 = n2 + n n n2 + 3n (giat n 3), kai n2 + 3n n2 + 2n + 1 = (n + 1)2. Ap tic prohgomenec sqseic prokptei to zhtomeno.
  12. 12. 12 1 Akraioi 3) 'Estw A na peperasmno snolo pou qei n stoiqea. Tte to pljoc twn uposunlwn tou A enai 2n. Qrhsimopoiome epagwg. Gia n = 0, h prtash alhjeei kajc to ken snolo qei 20 = 1 uposnola. Upojtoume tra ti h prtash alhjeei gia kje snolo me n N stoiqea. 'Estw ti to A qei n+1 stoiqea kai A. To snolo B = A {} qei n stoiqea kai sunepc to pljoc twn uposunlwn tou enai 2n. Ja metrsoume to pljoc twn uposunlwn tou A. 'Estw A. Diakrnoume do periptseic. 1) 'Estw / . Tte B kai sunepc to pljoc twn enai 2n. 2) 'Estw . Tte = {} me B, opte sumperanoume ti to pljoc twn enai pli 2n. Sunolik, uprqoun 2n + 2n = 2n+1 uposnola tou A, pou enai to zhtomeno. 4) Jerhma de Moivre. 'Estw R. Tte gia ton migadik arijm sun + ihm isqei (sun + ihm)n = sun(n) + ihm(n) gia kje n N. Gia n = 0 h apodeikta sqsh enai profanc. Upojtoume ti aut isqei gia n. Qrhsimopointac tic gnwstc trigwnometrikc tautthtec sun( + ) = sunsun hmhm kai hm( + ) = hmsun + hmsun, qoume (sun + ihm)n+1 = (sun + ihm)n (sun + ihm) = (sun(n) + ihm(n))(sun + ihm) = sun(n)sun hm(n)hm + i(sun(n)hm + hm(n)sun) = sun(n + ) + ihm(n + ) = sun((n + 1)) + ihm((n + 1)). Uprqei ma llh morf thc majhmatikc epagwgc pou enai qrsimh. 1.1.5 Jerhma (Deterh Morf thc Majhmatikc Epagwgc). 'Estw ti gia kje fusik arijm n dnetai mia prtash P(n), pou afor ton n, ttoia ste 1) h P(0) alhjeei, kai 2) gia kje n, an h P(k) alhjeei gia kje fusik arijm k me 0 k n, tte h P(n + 1) alhjeei. Tte h P(n) alhjeei gia kje fusik arijm n.
  13. 13. 1.1. Majhmatik Epagwg, Diwnumiko Suntelestc 13 Apdeixh. Arke na apodeiqje ti to snolo A = {n N|P(n) den alhjeei} enai ken. 'Estw ti to A den enai ken. Tte lgw tou Axi- matoc 1.1.1, to A periqei elqisto stoiqeo, stw m. Ap thn upjesh 1) qoume ti m 1. Ap ton orism tou m prokptei ti h prtash P(k) alh- jeei gia kje k m 1. Ap thn upjesh 2) prokptei ti h P(m) alhjeei, pou enai topo. Shmewsh. Sth Deterh Morf thc Majhmatikc epagwgc enai dunatn to arqik bma na mhn enai to 0 all opoiosdpote m N. Sgkrine me to Jerhma 1.1.3. Diwnumiko suntelestc Sth sunqeia ja asqolhjome me to diwnumik anptugma pou ja qrhsi- mopoihje sta epmena. 'Estw i n fusiko arijmo. Me n i sumbolzoume ton suntelest tou xi sto anptugma tou diwnmou (1 + x)n. Gia pardeigma, qoume 2 0 = 1, 2 1 = 2, 2 2 = 1, afo (1 + x)2 = 1 + 2x + x2. 'Omoia qoume 3 0 = 1, 3 1 = 3, 3 3 = 1, afo (1 + x)3 = 1 + 3x + 3x2 + x3. 'Etsi grfoume ex orismo (1 + x)n = n 0 + n 1 x + n 2 x2 + + n n xn . Sthn perptwsh pou qoume i > n sumfwnome ti n i = 0. Aut epitrpei kpoia omoiomorfa stic diatupseic protsewn pou aforon touc diwnumikoc suntelestc. (Bl. thn prth sqsh thc paraktw prtashc gia i = n + 1.) 1.1.6 Prtash. 1) 'Estw i, n fusiko arijmo me 1 i n + 1 . Tte n + 1 i = n i 1 + n i . 2) 'Estw i, n fusiko arijmo me i n . Tte n i = n! i!(n i)! (1) pou jtoume 0! = 1 kai gia kje jetik akraio n, n! = 1 2 n. Apdeixh. 1) Jewrome thn tautthta poluwnmwn (1 + x)n+1 = (1 + x)n (1 + x) = (1 + x)n + (1 + x)n x
  14. 14. 14 1 Akraioi kai sugkrnoume touc suntelestc tou xi sto arister kai dexi mloc. Sto arister mloc o en lgw suntelestc enai n + 1 i , en sto dexi enai n i + n i 1 . 2) Qrhsimopoiome epagwg sto n. Gia n = 0, opte i = 0, h apodeikta sqsh enai profanc. Upojtontac thn (1) ja apodexoume ti n + 1 i = (n + 1)! i!(n + 1 i)! . (2) An i = 0, ekola epalhjeetai h (2). 'Estw loipn 0 < i n + 1, opte ap to 1) thc prtashc kai thn isthta (1) qoume n + 1 i = n i 1 + n i = n! (i 1)!(n i + 1)! + n! i!(n i)! = n! (i 1)!(n i)! 1 n i + 1 + 1 i = n! (i 1)!(n i)! n + 1 (n i + 1)i = (n + 1)! i!(n + 1 i)! . kai ra h (2) isqei. 1.1.7 Efarmog. 'Estw A na peperasmno snolo me n stoiqea. Tte to pljoc twn uposun- lwn tou A pou qoun i stoiqea enai so me n i gia kje i me 0 i n. Efarmzoume epagwg sto n. An n = 0, tte i = 0 kai to zhtomeno enai profanc, afo to ken snolo qei akribc 0 0 = 1 uposnolo. 'Estw - ti kje peperasmno snolo me n stoiqea, n > 0, qei n i uposnola me i stoiqea. 'Estw ti to A qei n + 1 stoiqea. Ja metrsoume tra to pljoc twn uposunlwn tou A pou qoun i stoiqea. 'Estw a A kai B = A {}. 'Estw na uposnolo tou A pou qei i stoiqea. 'Opwc sto Pardeigma 1.1.4 3), diakrnoume do periptseic. 1) 'Estw / . Tte B. Ap thn upjesh thc epagwgc petai ti to pljoc twn enai n i . 2) 'Estw . Tte {} B. Ap thn upjesh thc epagwgc petai ti to pljoc twn
  15. 15. 1.1. Majhmatik Epagwg, Diwnumiko Suntelestc 15 {} (kai epomnwc to pljoc twn ) enai n i 1 . Sunolik uprqoun n i + n i 1 = n + 1 i uposnola tou A pou qoun i stoiqea. Askseic 1.1 1) Apodexte ti 1 1 2 + 1 2 3 + + 1 n(n + 1) = n n + 1 gia kje jetik akraio n. 2) Apodexte ti 13 + 23 + + n3 = 1 4 n2 (n + 1)2 = (1 + 2 + + n)2 gia kje jetik akraio n. 3) Apodexte ti i) 3n < n3 gia kje n N, n 2. ii) mn < nm gia kje m, n N me m 3 kai n 2. Updeixh: epagwg sto m. 4) Apodexte ti gia kje k, m, n N isqoun oi paraktw tautthtec. i) n 0 n 1 + n 2 + (1)n n n = 0, ii) n 0 + n 1 + n 2 + + n n = 2n , iii) 2n n = n 0 2 + n 1 2 + + n n 2 , iii) m + n k = m 0 n k + m 1 n k 1 + + m k n 0 . Updeixh: Gia thn prth tautthta qrhsimopoiste ti (1 + (1))n = 0. Gia thn trth, sugkrnete touc suntelestc tou xn sthn tautthta (1 + x)2n = (1 + x)n(1 + x)n. 5) 'Estw f0 = 0, f1 = 1 kai fn = fn1 + fn2 gia n 2 (akolouja tou Fibonacci). Apodexte ti gia kje n N isqoun: i) f0 + f1 + + fn = fn+2 1. ii) fn+2fn f2 n+1 = (1)n+1.
  16. 16. 16 1 Akraioi iii) fn = 1 5 (n n ), pou = 1 + 5 2 kai = 1 5 2 (oi , ikanopoion thn exswsh x2 x 1 = 0). iv) f2n = n 0 f0 + n 1 f1 + + n n fn. Updeixh: Qrhsimopoiste to iii). 6) Gia kje r, n N me r n apodexte ti r r + r + 1 r + + n r = n + 1 r + 1 . 7) 'Estw R. Tte gia ton migadik arijm sun + ihm isqei (sun + ihm)n = sun(n) + ihm(n) gia kje n Z. 8) Apodexte ti ta aklouja enai isodnama. i) Axwma 1.1.1 (Axwma Elaqstou). ii) Jerhma 1.1.2 (Majhmatik Epagwg) iii) Jerhma 1.1.5 (Deterh morf thc Majhmatikc Epagwgc).
  17. 17. 1.2. Diairetthta 17 1.2 Diairetthta Sthn Pargrafo aut ja meletsoume thn nnoia thc diairetthtac sto Z. Afo apodexoume ton Algrijmo Diareshc, ja anaptxoume thn nnoia tou mgistou koino diairth me th bojeia tou opoou ja apodexoume to Jeme- lidec Jerhma thc Arijmhtikc, smfwna me to opoo kje akraioc arijmc diforoc twn 0 kai 1 grfetai wc ginmeno prtwn kat trpo ousiastik monadik. Diairetthta kai prtoi arijmo 'Estw a, b Z. Ja lme ti o a diaire ton b ( ti o a enai diairthc tou b ti o b enai pollaplsio tou a) an uprqei c Z me b = ac. Ja grfoume tte a|b. Parathrome ti kje akraioc enai diairthc tou 0 en to 0 enai diairthc mno tou eauto tou. 'Estw a, b, c treic akraioi arijmo. Ekola diapistnoume ti isqoun oi paraktw idithtec, tic opoec ja qrhsimopoiome sth sunqeia qwrc idiaterh mnea. An a|b kai a|c, tte a|bx + cy gia kje x, y Z. Idiatera, a|b c. An a|b kai b|a, tte a = b. An a|b kai b|c, tte a|c An a|b kai oi a, b enai jetiko, tte a b. Endeiktik apodeiknoume thn prth idithta: ap thn upjesh, uprqoun akraioi e, f ttoioi ste b = ae kai c = af. Antikajistntac qoume ti bx + cy = aex + afy = a(ex + fy). Sunepc, a|bx + cy. 'Enac jetikc akraioc p = 1 lgetai prtoc arijmc an oi mnoi diairtec tou enai oi 1 kai p. Gia pardeigma, ap touc 1, 2, 3, 4, 5, 6, 7, 8 kai 9 oi prtoi arijmo enai oi 2, 3, 5 kai 7. O krioc skopc mac sthn Pargrafo aut enai na apodexoume ti kje akraioc arijmc diforoc twn 0, 1 grfetai wc ginmeno prtwn kat trpo ousiastik monadik (Jemelidec Jerhma thc Arijmhtikc). Antstoiqo apotlesma ja sunantsoume sthn Enthta 2, tan meletsoume polunuma. Epshc h ida thc monadikc paragontopohshc anaptssetai pio genik sthn Enthta 3. Sthn diatpwsh thc akloujhc Prtashc deqmaste ti kje prtoc arij- mc enai ginmeno prtwn arijmn kat tetrimmno trpo. 1.2.1 Prtash. Kje jetikc akraioc diforoc tou 1 enai ginmeno prtwn arijmn.
  18. 18. 18 1 Akraioi Apdeixh. 'Estw ti h prtash den isqei kai stw M to snolo twn ake- rawn n > 1 oi opooi den enai ginmena prtwn arijmn. Tte M = kai ap to Axwma 1.1.1 uprqei elqisto stoiqeo m M. 'Eqoume m > 1 kai o m den enai prtoc (afo kje prtoc enai kat tetrimmno trpo ginmeno prtwn). Sunepc m = ab gia kpoiouc akeraouc a, b pou 1 < a < m kai 1 < b < m. Ap ton orism tou m prokptei ti a / M kai b / M, dhlad oi a, b enai ginmena prtwn arijmn. Tte mwc to dio sumbanei gia to ginmen touc m = ab, dhlad m / M. Aut enai topo. Sunepc blpoume ti kje akraioc diforoc twn 0,1 grfetai sthn mor- f p1 . . . pk, pou oi pi enai prtoi arijmo (qi anagkastik diakekrimnoi). 1.2.2 Jerhma (Eukledhc). 1 Uprqoun peiroi prtoi arijmo. Apdeixh. 'Estw ti to snolo twn prtwn arijmn enai peperasmno kai ti enai to {p1, . . . , pm}. O akraioc p1p2 . . . pm + 1 qei nan prto diai- rth smfwna me thn Prtash 1.2.1, stw pi. Afo pi|p1p2 . . . pm + 1 kai pi|p1p2 . . . pm sumperanoume ti o pi diaire th diafor, dhlad pi|1. Aut enai topo. To epmeno apotlesma perigrfei mia ap tic pio shmantikc idithtec tou Z kai ja qrhsimopoihje pollc forc sta paraktw. 1.2.3 Jerhma (Algrijmoc Diareshc Eukledeia Diaresh). 'Estw a, b Z me a > 0. Tte uprqoun monadiko q, r Z me tic idithtec b = qa + r kai 0 r < a. Apdeixh. 1)'Uparxh: Jtoume M = {b ta 0|t Z}. Isqei M = (arke na jewrsoume t < 0 me |t| arket meglo, pou me |t| sumbolzoume thn apluth tim tou t), opte ap to Axwma 1.1.1 uprqei elqisto stoiqeo r M. 'Eqoume r = b qa, gia kpoion akraio q, kai ja dexoume ti r < a. An sque r a, tte antikajistntac ja eqame b (q + 1)a 0, opte b (q + 1)a M. Aut enai topo ap ton orism tou r, giat r > r a = b (q + 1)a. 2) Monadikthta: 'Estw ti eqame b = qa + r, 0 r < a, pou q, r Z kai b = q a + r , 0 r < a, pou q , r Z. Tte lambnoume (q q )a = r r 1 H apdeixh aut, pwc kai o Eukledeioc Algrijmoc, periqontai sto rgo Stoiqea tou Eukledh.
  19. 19. 1.2. Diairetthta 19 kai a < r r < a. 'Ara a < (q q )a < a kai sunepc (afo a > 0) 1 < q q < 1. Epeid q q Z, sumperanoume ti q q = 0. Sunepc r r = (q q )a = 0.
  20. 20. 20 1 Akraioi Shmeiseic 1. H apdeixh thc parxhc twn q kai r sto prohgomeno Jerhma sunstatai sthn austhr diatpwsh thc akloujhc aplc idac. Ac upojsoume gia eukola ti b > 0. Afairome ap ton b ta mh arnhtik pollaplsia tou a, dhlad ta 0, a, 2a, 3a, . . ., tsi ste h diafor b ta na paramnei mh arnhtik. To teleutao ttoio pollaplsio enai to qa kai h diafor b qa enai to r. 2. An paraleyoume to a > 0 ap thn upjesh tou prohgomenou Jewr- matoc kai jewrsoume a = 0, tte to mno pou allzei enai h anisthta tou sumpersmatoc pou prpei na enai tra 0 r < |a|. (H apdeixh petai amswc ap to Jerhma: an o a enai arnhtikc, efarmzoume to Jerhma gia a sth jsh tou a). 3. O fusikc arijmc r tou algorjmou diareshc onomzetai to uploipo thc diareshc tou b me to a. Mgistoc koinc diairthc 'Estw a, b Z me toulqiston nan diforo tou mhdenc. 'Enac mgistoc koinc diairthc twn a kai b (sumbolik (a, b) (a, b)) enai nac jetikc akraioc d pou qei tic idithtec 1. d|a kai d|b 2. an c Z me c|a kai c|b, tte c|d. Gia pardeigma, qoume (12, 30) = 6. Parathrome ti an d = (a, b), tte kje lloc koinc diairthc c twn a kai b enai ttoioc ste c d lgw thc sunjkhc 2 tou orismo. 'Etsi exhgetai h onomasa mgistoc koinc diairthc. Sto stdio aut den enai telewc safc ti uprqei gia kje a, b Z me toulqiston nan diforo tou mhdenc. Pra ap thn parxh, to paraktw apotlesma parqei ma shmantik parstash tou pou ja qrhsimopoihje suqn sta paraktw. 1.2.4 Jerhma. 'Estw a, b Z me toulqiston nan diforo tou mhdenc. Tte uprqei monadikc mgistoc koinc diairthc twn a kai b. Epiplon upr- qoun x, y Z ttoioi ste (a, b) = ax + by. Apdeixh. 'Uparxh: Jtoume M = {ax + by|x, y Z kai ax + by > 0} kai parathrome ti M = (giat a2 + b2 > 0). 'Estw d to elqisto stoiqeo tou M, to opoo uprqei smfwna me to Axwma 1.1.1. 'Eqoume d = ax + by gia kpoiouc x, y Z.
  21. 21. 1.2. Diairetthta 21 Ja apodexoume ti d = (a, b). Gia ton skop aut, deqnoume prta ti d|a kai d|b. Ap to Jerhma 1.2.3, uprqoun akraioi q, r ttoioi ste a = qd+r, 0 r < d. 'Eqoume r = aqd = aq(ax+by) = a(1qx)+b(qy). Aut shmanei ti an r = 0, tte r M, pou mwc enai topo lgw tou elaqstou tou d. 'Ara r = 0 kai sunepc d|a. 'Omoia apodeiknetai ti d|b. Tra stw c|a kai c|b. Epeid qoume d = ax + by sumperanoume ti c|d. 'Ara prgmati o d enai nac twn a kai b. Monadikthta: An d kai d tan mgistoi koino diairtec twn a, b, ja eqame d|d (giat o d enai nac twn a, b) kai d |d (giat o d enai nac twn a, b). 'Ara d = d , kai afo d, d > 0 parnoume d = d . H prohgomenh apdeixh parqei nan trpo prosdiorismo tou (wc to elqisto tou sunlou M). Uprqei mwc nac pio praktikc trpoc (Eu- kledeioc Algrijmoc) pou perigrfetai paraktw. Prin ap aut, mwc, ja apodexoume to Jemelidec Jerhma thc Arijmhtikc. Gia to skop aut qreiazmaste to akloujo Lmma, pou petai ap to Jerhma 1.2.4. 1.2.5 Lmma. 'Estw a, b, p Z pou o p enai prtoc arijmc. An o p diaire to ginmeno ab, tte o p diaire toulqiston nan ap touc a kai b. Apdeixh. 'Estw ti o p den diaire ton a. Afo o p enai prtoc qoume (a, p) = 1. Smfwna me to Jerhma 1.2.4 uprqoun akraioi x kai y ttoioi ste 1 = ax + py. 'Ara b = abx + pyb. Epeid p|abx kai p|pyb prokptei ti p|abx + pyb, dhlad p|b. 1.2.6 Parathrseic. 1. Me epagwg mpore na genikeuje to prohgomeno Lmma sthn perptwsh perissotrwn paragntwn: An o p enai nac prtoc arijmc pou diaire to ginmeno a1 . . . am, (ai Z), tte ja diaire toulqiston nan ap touc ai. H apdeixh afnetai san skhsh. 2. To Lmma 1.2.5 enai idiatera qrsimo. Mia tupik efarmog tou enai h apdeixh ti o arijmc 2 enai rrhtoc. Prgmati, stw ti 2 = m n , pou oi m, n enai jetiko akraioi. Mporome na upojsoume ti (m, n) = 1, giat diaforetik aplopoiome to klsma. 'Eqoume m2 = 2n2. An n = 1, tte smfwna me thn Prtash 1.2.1 uprqei prtoc p me p|n. Tte p|m2 kai ra p|m, ap to Lmma 1.2.5. Sunepc p|(m, n), dhlad p|1, pou enai topo. 'Ara n = 1. All tte 2 = m Z, pou enai topo. Me parmoio trpo mpore na apodeiqje ti gia kje jetik akraio a pou den enai tetrgwno akeraou o arijmc a enai rrhtoc ('Askhsh 15).
  22. 22. 22 1 Akraioi 1.2.7 Jerhma (Jemelidec Jerhma thc Arijmhtikc). Kje ak- raioc a > 1 grfetai wc ginmeno prtwn arijmn, a = p1 . . . pm, pi prtoc. H parstash aut enai monadik me thn exc nnoia: an a = p1 . . . pm = q1 . . . qn (pi, qj prtoi), tte m = n kai, met endeqomnwc ap kpoia anaditaxh, qou- me p1 = q1, . . . , pm = qm. Apdeixh. 'Estw a > 1 nac akraioc arijmc. Sthn Prtash 1.2.1 dexame ti uprqoun prtoi arijmo p1, . . . , pm pou qoun thn idithta a = p1 . . . pm. Ja dexoume tra th monadikthta thc parstashc autc. 'Estw ti a = p1 . . . pm = q1 . . . qn, pou oi pi, qj enai prtoi arijmo. Mporome na upoj- soume ti m n, opte efarmzoume epagwg sto m. Gia m = 1 qoume p1 = q1 . . . qn opte o p1 ja diaire kpoion ap touc qj smfwna me thn Para- trhsh 1.2.6, stw ton q1. Epeid o q1 enai prtoc parnoume p1 = q1. 'Etsi 1 = q2 . . . qn, prgma pou shmanei ti n = 1. 'Estw tra m > 1. Ap thn isthta p1 . . . pm = q1 . . . qn parnoume pwc prin, met endeqomnwc ap mia anaditaxh twn qj, ti p2 . . . pm = q2 . . . qn. To zhtomeno prokptei tra ap thn epagwgik upjesh. Smfwna me to prohgomeno Jerhma, kje akraioc a = 0, 1 qei mo- nadik parstash (qwrc na lambnetai upyh h seir twn paragntwn) thc morfc a = pa1 1 . . . pan n , pou oi pi enai an do diforoi prtoi arijmo kai oi ai enai jetiko akraioi. H paragontopohsh aut kaletai anlush tou a se ginmeno prtwn kai oi prtoi arijmo p1, . . . , pn onomzontai prtoi pargontec tou a. Do akraioi a,b onomzontai sqetik prtoi an (a, b) = 1. Isodna- ma, do akraioi enai sqetik prtoi an den qoun koin prto pargonta. Eukledeioc Algrijmoc Perigrfoume tra ma praktik diadikasa pou upologzei to (a, b) kai epiplon prosdiorzei akeraouc x, y tsi ste na isqei (a, b) = ax + by smfwna me to Jerhma 1.2.4. Onomzetai de Eukledeioc algrijmoc kai sthrzetai sthn epanalambanmenh efarmog thc akloujhc aplc paratrh- shc: b = aq + r (a, b) = (r, a). (1) Prgmati, ap thn isthta b = aq + r petai ti (a, b)|r. 'Eqoume dhlad (a, b)|r kai (a, b)|a, opte ap ton orism tou parnoume (a, b)|(r, a). Me parmoio trpo apodeiknetai ti (r, a)|(a, b). Epeid o enai jetikc akraioc, ap tic do teleutaec sqseic prokptei ti (a, b) = (r, a).
  23. 23. 1.2. Diairetthta 23 Pardeigma 'Estw a = 50 kai b = 240. Ap thn tautthta diareshc lambnoume diadoqik 240 = 4 50 + 40 50 = 1 40 + 10 40 = 4 10 + 0 Efarmzontac thn (1) se kje mia ap tic treic isthtec, qoume (50, 240) = (40, 50) = (10, 40) = (0, 10) = 10. Gia na prosdiorsoume akeraouc x, y ttoiouc ste 10 = 50x + 240y xekinme ap thn teleutaa tautthta diareshc pou qei mh mhdenik uploipo (thn 50 = 1 40 + 10) kai ektelome diadoqikc antikatastseic ergazmenoi proc ta pnw. (H diadikasa aut onomzetai antanaresh). 10 = 50 1 40 = = 50 1 (240 4 50) = = 50 5 + 240 (1), opte x = 5 kai y = 1. Ap to prohgomeno pardeigma blpoume ti o do akerawn a kai b(a = 0) isotai me to teleutao mh mhdenik uploipo (rn) pou sunantme ston antstoiqo Eukledeio algrijmo: b = aq + r, 0 r < a a = rq1 + r1, 0 r1 < r r = r1q2 + r2, 0 r2 < r1 r1 = r2q3 + r3, 0 r3 < r2 rn2 = rn1qn + rn, 0 rn < rn1 rn1 = rnqn+1 + 0. Prgmati, efarmzontac diadoqik thn (1) qoume (a, b) = (r, a) = (r1, r) = (r2, r1) = = (rn, rn1) = (0, rn) = rn.
  24. 24. 24 1 Akraioi Ed uprqei to erthma an h diadikasa twn diadoqikn diairsewn termatzetai met ap peperasmno arijm epanalyewn. H apnthsh enai jetik, giat ta uploipa sqhmatzoun mia austhr fjnousa akolouja fusikn arijmn me arq to a, dhlad qoume a > r > r1 > > rn . Gia na gryoume ton (a, b) sth morf ax + by prta epiloume tic parapnw exisseic wc proc ta uploipa lambnontac r = b aq r1 = a rq1 r2 = r r1q2 r3 = r1 r2q3 rn2 = rn4 rn3qn2 rn1 = rn3 rn2qn1 rn = rn2 rn1qn. Sth sunqeia antikajistome sthn teleutaa exswsh ton rn1 ap thn prote- leutaa rn = rn2 (rn3 rn2qn1)qn = rn2(1 + qn1qn) + rn3(qn). Sth na exswsh antikajistome thn kfrash pou qoume gia ton rn2. rn = rn2(1+qn1qn)+rn3(qn) = (rn4rn3qn2)(1+qn1qn)+rn3(qn). Suneqzontac kat ton trpo aut fjnoume telik se mia parstash thc morfc rn = ax + by. Parathrome ti h parapnw mjodoc parqei mia na apdeixh thc parxhc tou (a, b) kai thc parxhc x, y Z pou qoun thn idithta (a, b) = ax + by. Shmeinoume ti oi x, y den enai anagkastik monadiko , afo ax + by = a(x + b) + b(y a). Isqei mwc (x, y) = 1 ('Askhsh 9). Parathrseic sthn anlush akerawn se ginmeno prtwn 1) H anlush akerawn se ginmeno prtwn parqei nan llo trpo upo- logismo tou . 'Estw a, b jetiko akraioi kai a = pa1 1 pan n kai b = pb1 1 pbn n (2) pou pi enai an do diforoi prtoi kai ai, bi N. (Kai oi do paragontopoi- seic periqoun touc diouc prtouc p1, . . . , pn giat epitrpoume ed mhdenikoc
  25. 25. 1.2. Diairetthta 25 ekjtec). Parathrome ti isqei a|b ai bi gia kje i. (3) Prgmati, h apdeixh thc katejunshc enai mesh. Antstrofa, stw a|b kai stw ti a1 > b1. Tte ap thn (2) parnoume ti o pa1b1 1 pa2 2 . . . pan n diaire ton pb2 2 . . . pbn n . Ap thn Paratrhsh 1.2.6 1. blpoume ti p1|pi gia kpoio i = 1. Aut enai topo. Qrhsimopointac thn (3) kai ton orism tou mporome na dome ti (a, b) = pd1 1 . . . pdn n , pou gia kje i enai di = min{ai, bi}. (4) Gia pardeigma, an a = 25 34 50 kai b = 2 36 52, tte (a, b) = 2 34 50. Ap thn (4) mporome na sungoume qrsimec sqseic, pwc gia pardeigma thn (ca, cb) = c (a, b). (5) Shmeinoume ti o trpoc upologismo tou pou dnetai sthn (4) den enai pol praktikc gia meglouc arijmoc giat propojtei th gnsh anal- sewn se ginmena prtwn. Genik h eresh thc anlushc enc meglou arijmo se ginmeno prtwn enai nac qronobroc upologismc pou pollc forc kaj- statai praktik adnatoc, akma kai an qrhsimopoihjon isquro upologistc. Se aut akribc to gegonc sthrzetai mia axipisth kai diadedomnh mjodoc kruptogrfhshc mhnumtwn, h RSA. Thn mjodo aut parousizoume suno- ptik sthn Pargrafo 1.6 paraktw. 2) 'Estw a, b akraioi ap touc opoouc toulqiston nac enai mh mhdenikc. 'Ena elqisto koin pollaplsio ( ) twn a, b enai nac jetikc akraioc e pou qei tic idithtec a|e kai b|e an o c Z enai ttoioc ste a|c kai b|c, tte e|c. Enai safc ti an uprqei twn a, b tte aut enai monadik. Ac jewr- soume tic paragontopoiseic twn a, b se ginmena prtwn, a = pa1 1 . . . pan n kai b = pb1 1 . . . pbn n . Gia kje i jtoume ei = max{ai, bi} kai orzoume ton jetik akraio e = pe1 1 . . . pen n . Gia pardeigma, an a = 25 34 50 kai b = 2 36 52, tte e = 25 36 52. Qrhsimopointac thn (3), blpoume ti o e ikanopoie tic do sunjkec ston orism tou ekp. Sunepc to twn a, b uprqei kai enai monadik. Sumbolzetai de me (a, b).
  26. 26. 26 1 Akraioi Ap th deterh idithta ston orism tou enai safc ti anmesa stouc jetikoc akeraouc pou enai koin pollaplsia twn a kai b to (a, b) enai o elqistoc. Qrhsimopointac th sqsh min{ai, bi} + max{ai, bi} = ai + bi, ekola a- podeiknetai ti (a, b) (a, b) = |ab|. (6) 1.2.8 Paradegmata. 1. 'Estw a, b, c Z me a|bc. An isqei (a, b) = 1, tte a|c. (Sgkrine me to Lmma 1.2.5). Epeid isqei (a, b) = 1, uprqoun x, y Z me 1 = ax + by (Jerhma 1.2.4). Parnoume c = cax + cby. 'Eqoume a|cax kai ap thn upjesh petai ti a|cby. 'Ara o a ja diaire ton cax+cby = c. (Mia llh apdeixh mpore na doje qrhsimopointac analseic se ginmena prtwn). 2. 'Estw a, m, n Z me m|a kai n|a. An (m, n) = 1, tte mn|a. Epeid qoume m|a kai n|a, parnoume e|a, pou e = (m, n). All ap thn (6) qoume (m, n) = |mn|, giat (m, n) = 1. 3. An a, b Z kai (a, b) = d, tte (a/d, b/d) = 1. An o akraioc c diaire kai ton a/d kai ton b/d, tte o cd diaire kai ton a kai ton b. 'Ara o cd ja diaire ton (a, b) = d pou shmanei ti c = 1. 4. An to ginmeno do sqetik prtwn jetikn akerawn arijmn a, b enai tetrgwno akeraou, tte oi a, b enai tetrgwna akerawn. 'Estw ab = c2 kai a = pa1 1 . . . par r , b = pb1 1 . . . pbr r , c = pc1 1 . . . pcr r analseic se ginmena prtwn pwc sthn (2). Epeid (a, b) = 1, blpoume ti gia kje i to pol nac ap touc ai, bi enai mh mhdenikc. Ap th sqsh (pa1 1 . . . par r )(pb1 1 . . . pbr r ) = p2c1 1 . . . p2cr r kai th monadikthta thc paragontopohshc sto Z qoume ti ai + bi = 2ci gia kje i. Sunepc kje ai (kai bi) enai so ete me 0 ete me 2ci. Epomnwc o a (kai o b) enai tetrgwno akeraou. 5. Na breje o (n6 1, n10 1) Ap tic sqseic n10 1 = n4 (n6 1) + n4 1, n6 1 = n2 (n4 1) + n2 1, n4 1 = (n2 + 1)(n2 1),
  27. 27. 1.2. Diairetthta 27 sungoume ti (n10 1, n6 1) = (n6 1, n4 1) = (n4 1, n2 1) = n2 1. Genik isqei (na 1, nb 1) = nd 1, pou d = (a, b) ('Askhsh 17). 6. 'Estw (m, n) = 1. Na brejon oi dunatc timc gia ton (m + n, m n). Ja dexoume ti h apnthsh enai 1 2. 'Estw d = (m + n, m n). Epeid d|m + n kai d|m n, parnoume d|(m + n) + (m n) kai d|(m + n) (m n), dhlad d|2m kai d|2n. 'Ara d|(2m, 2n). 'Omwc (2m, 2n) = 2(m, n) ap thn (??) kai ra d|2, dhlad d = 1 2. Apodexame ti oi pijanc timc tou d enai 1 kai 2. Gia m = 2, n = 1 qoume d = 1, en gia m = 3, n = 1 qoume d = 2. Sunepc oi dunatc timc tou d enai 1 2. 7. 'Estw a, b, n jetiko akraioi. Tte an|bn an kai mno an a|b. 'Estw a = pa1 1 . . . par r kai b = pb1 1 . . . pbr r pou pi enai an do diforoi prtoi arijmo kai ai, bi N. 'Eqoume an = pna1 1 . . . pnar r , bn = pnb1 1 . . . pnbr r . Ap th sqsh (3) parnoume an |bn nai nbi gia kje i ai bi gia kje i a|b. 8. Na brejon loi oi jetiko akraioi m, n ttoioi ste mn = nm. Ja dexoume ti ta zhtomena zegh (m, n) enai ta exc: (2,4), (4,2) kai (m, m) pou m enai jetikc akraoc. Mporome na upojsoume ti m n 2. Tte nn|nm dhlad nn|mn. Ap thn prohgomenh Efarmog parnoume n|m. 'Estw m = an. Tte antikajistntac sthn arqik exswsh kai lambnontac n-stec rzec parnoume an = na. 'Omwc enai ekolo na deiqje me epagwg sto a ti an < na gia kje a 3 kai n 2 ('Askhsh 1.1.3). Sunepc a = 1 2. Gia a = 1 qoume m = n kai gia a = 2 qoume 2n = n2, opte n = 2. 9) 'Estw b nac akraioc me b > 1. Tte kje jetikc akraioc n qei monadik parstash thc morfc n = akbk + ak1bk1 + + a1b + a0,
  28. 28. 28 1 Akraioi pou k, aj N me 0 aj b 1 gia j = 0, 1, . . . , k kai ak = 0. Prgmati, efarmzontac diadoqik ton Algrijmo Diareshc qoume n = bq0 + a0, 0 a0 b 1 q0 = bq1 + a1, 0 a1 b 1 ... qk2 = bqk1 + ak1, 0 ak1 b 1 qk1 = b0 + ak, 0 < ak b 1. 'Eqoume q0 > q1 > ... . Upojtoume ti qk enai to prto phlko pou isotai me 0. Sthn prth isthta n = bq0 + a0 antikajistome to q0 ap th deterh, sth sunqeia to q1 ap thn trth kai otw kaj' exc. Telik prokptei ma parstash thc morfc n = akbk +ak1bk1 + +a1b+a0, pou 0 aj b 1 gia j = 0, 1, . . . , k kai ak = 0. Gia thn apdeixh thc monadikthtac qrhsimopoiome th deterh morf thc Majhmatikc Epagwgc. To zhtomeno enai profanc gia n = 1. 'Estw n > 1. Upojtoume ti h monadikthta isqei gia kje jetik akraio mikrtero tou n. 'Estw ti qoume kai thn parstash n = clbl + cl1bl1 + + c1b + c0, pou l, cj N me 0 cj b 1 gia j = 0, 1, . . . , l kai cl = 0. 'Eqoume a0 = c0, giat kajna ap aut enai to uploipo thc diareshc tou n me ton b. To zhtomeno prokptei an efarmoste h upjesh thc epagwgc ston akraio (n a0)/b. H parapnw isthta pou apodexame onomzetai h parstash tou n wc proc th bsh b. Gia b = 10 qoume th sunjh dekadik parstash tou n. Gia b = 2 qoume th duadik parstash tou n. Epishmanoume ti h apdeixh thc parxhc pou dsame parqei nan algrijmo me ton opoo mporome na brome ta 'yhfa' ai sthn parstash tou n wc proc mia bsh b. Askseic 1.2 1) An o p enai prtoc arijmc me p|an, apodexte ti pn|an. Updeixh: Lmma 1.2.5 2) An o p enai prtoc arijmc me p|a kai p|a2 + b2, apodexte ti p|b. 3) Prosdiorste ton (36, 210), pwc kai akeraouc x, y ttoiouc ste (36, 210) = 36x + 210y.
  29. 29. 1.2. Diairetthta 29 4) Na breje o akraioc a > 1 an (a, a + 3) = a. 5) Apodexte ti (m, n) = (m + kn, n) gia kje k N. 6) Apodexte ti 6|a an kai mno an (a, a + 2) = 1 kai (a, a + 3) = 1. 7) Apodexte ti (m + n, mn) = 1 an (m, n) = 1. 8) Apodexte ti gia kje n N isqei (3n + 1, 10n + 3) = 1. 9) An d = (m, n) kai d = mx + ny, apodexte ti (x, y) = 1. 10) Apodexte th sqsh (6). 11) 'Estw a, m Z me m > 0 kai a = 1. Apodexte ti am 1 a 1 , a 1 = (a 1, m). Updeixh: am 1 a 1 = (am1 1) + (am2 1) + + (a 1) + m. 12) An a1, . . . , an Z (qi loi mhdn) orzoume ton (a1, . . . , an) wc nan jetik akraio d pou qei tic idithtec 1) d|ai gia kje i, kai 2) an c Z me c|ai gia kje i tte c|d. i) Apodexte ti o (a1, . . . , an) uprqei, enai monadikc kai epiplon uprqoun xi Z me (a1, . . . , an) = a1x1 + + anxn. ii) Apodexte ti an ai = 0 gia kje i kai n 3 tte (a1, . . . , an) = (a1, . . . , an2, (an1, an)) = (a1, (a2, . . . , an)). iii) Upologste ton (135, 170, 205, 310). iv) Genikeste th sqsh (4). 13) An a1, . . . , an Z enai mh mhdeniko akraioi, orzoume to (a1, . . . , an) wc nan jetik akraio e pou qei tic idithtec 1) ai|e gia kje i, kai 2) an c Z me ai|c gia kje i tte e|c. i) Apodexte ti to (a1, . . . , an) uprqei, enai monadik kai anmesa sta jetik koin pollaplsia twn a1, . . . , an enai to elqisto. ii) An ta ai enai mh mhdenik kai n 3, apodexte ti (a1, . . . , an) = (a1, . . . , an2, (an1, an)) = (a1, (a2, . . . , an)). iii) Apodexte ti (a1, a2, a3) = a1a2a3(a1, a2, a3) (a1, a2)(a2, a3)(a1, a3) Updeixh: Gia kje arijmoc a, b, c o max{a, b, c} enai soc me a + b + c min{a, b} min{b, c} min{a, c} + min{a, b, c}.
  30. 30. 30 1 Akraioi 14) Prosdiorste louc touc jetikoc akeraouc m, n ste (m, n) = 100. 15) Apodexte ti gia kje jetik akraio a pou den enai tetrgwno akeraou to a enai rrhtoc. 16) Poi enai ta elqista stoiqea twn paraktw sunlwn; i) {24a + 36b > 0|a, b Z} ii) {24a + 36b + 8c > 0|a, b, c Z} iii) {a > 0|a enai pollaplsio tou 24 kai tou 36}. 17) Apodexte ti (na 1, nb 1) = nd 1, pou d = (a, b), a, b, n enai jetiko akraioi, n > 1. 18) 'Estw a, b Z, a > 1. Apodexte ti uprqoun monadik q, r Z me b = qa + r kai a/2 r < a/2. 19) i) Apodexte ti kje prtoc arijmc diforoc tou 2 enai thc morfc 4n + 1 4n + 3, n N. ii) Apodexte ti kje fusikc arijmc thc morfc 4n + 3 qei nan toulqiston prto diairth thc morfc 4n + 3. iii) Apodexte ti uprqoun peiroi prtoi arijmo thc morfc 4n + 3, pou n N. Updeixh: Tropopoiste katllhla thn apdeixh tou Eukledh ti u- prqoun peiroi prtoi arijmo jewrntac ton arijm 4a1 . . . am +3, pou {3, a1, . . . , am} enai to snolo twn prtwn thc morfc 4n+3. Shmewsh: 'Ena fhmismno kai dskolo jerhma thc Jewrac Arij- mn enai aut tou Dirichlet, pou lei ti se kje arijmhtik prodo an+b, n N, pou (a, b) = 1, uprqoun peiroi prtoi arijmo. 20) An a, n enai jetiko akraioi, ttoioi ste n > 1 kai o an1 enai prtoc, apodexte ti a = 2 kai o n enai prtoc. 21) Apodexte ti gia kje n N o akraioc 52n+1 +62n+1 enai pollaplsio tou 11. 22) Na brejon loi oi prtoi p ste o p + 5 na enai prtoc. 23) Apodexte ti uprqei monadik trida thc morfc (p, p + 2, p + 4), pou oi p, p + 2, p + 4 enai prtoi arijmo. Shmewsh. Paramnei mqri smera anoikt to erthma an uprqoun peira zegh thc morfc (p, p + 2), pou oi p, p + 2 enai prtoi arijmo.
  31. 31. 1.2. Diairetthta 31 24) H skhsh aut dnei na pardeigma uposunlou tou N pou, en kje stoiqeo grfetai wc ginmeno prtwn, h graf den enai monadik, kai tsi den isqei se aut to anlogo Jemelidec Jerhma thc Arijmhtikc. 'Estw 2Z to snolo twn artwn akerawn. 'Ena stoiqeo q tou 2Z onom- zetai prto, an den uprqoun a, b 2Z me q = ab. Gia pardeigma, ta 2, 6, 10, 30 enai prta stoiqea tou 2Z. Apodexte ti kje mh mhdenik stoiqeo tou 2Z grfetai wc ginmeno prtwn stoiqewn. Parathrste ti to 60 = 2 30 = 6 10 grfetai kat do diaforetikoc trpouc wc ginmeno prtwn stoiqewn tou 2Z. 25) An (m, n) = 1, poic enai oi dunatc timc gia ton (m2+n2, m+n); 26) Exetste poic ap tic paraktw sunepagwgc enai swstc. 'Estw a, b Z kai n N. a|bn a|b an|bn a|b an|b a|b a3|b2 a|b 27) 'Estw a, m, n jetiko akraioi me m < n. i) Apodexte ti a2m + 1|a2n 1. ii) Apodexte ti (a2m + 1, a2n + 1) = 1 2. iii) Qrhsimopointac to ii) apodexte ti uprqoun peiroi prtoi. 28) i) Apodexte ti h exswsh x2 y2 = 2 den qei lsh me x, y Z. ii) Lste thn exswsh 1 x + 1 y = 1 7 pou x, y Z. (Updeixh: Paragontopoiste). 29) Dexte ti gia kje n N uprqei p prtoc me n < p n! + 1 kai kat sunpeia uprqoun peiroi prtoi. 30) Dexte ti gia gia kje n N, n 2, den uprqei prtoc arijmc p me n! + 2 p n! + n. 31) Gia thn akolouja Fibonacci ('Askhsh 1.1.5) dexte ti (fn, fn+1) = 1 gia kje n N. 32) Gia thn akolouja Fibonacci ('Askhsh 1.1.5) apodexte ti to fn diairetai me to 3 an kai mno an to n diairetai me to 4.
  32. 32. 32 1 Akraioi 33) 'Estw m, n do jetiko akraioi. Apodexte ti (m, n) = (m + n, (m, n)). 34) Apodexete ti gia kje akraio n > 1, o rhtc arijmc 1 + 1/2 + 1/3 + + 1/n den enai akraioc. Updeixh: 'Estw ti 1+1/2+1/3+ +1/n = q N. 'Estw 2 h mgisth dnamh tou 2 pou enai mikrterh sh ap to n. 'Estw r to ginmeno twn mgistwn dunmewn twn perittn prtwn pou enai mikrterec sec ap to n. Pollaplasiste me 21r gia na fjsete se topo. 35) 'Estw f(x) = a0 + + anxn na polunumo, pou ai Z kai n 1. An na toulqiston ap ta ai den enai mhdn, dexte ti uprqei y Z ttoioc ste o f(y) den enai prtoc.
  33. 33. 1.3. Isotimec 33 1.3 Isotimec Suqn sumbanei oi lseic problhmtwn pou aforon akeraouc na exartntai mno ap uploipa diairsewn. Ac jewrsoume na pol apl pardeigma. H apnthsh sto erthma poi hmra thc edbomdac ja enai 7001 hmrec ap thn epmenh Kuriak fanetai amswc an skefjome ti oi hmrec thc ebdomdac epanalambnontai me perodo 7 kai ti 7001 = 71000+1. Sunepc h apnthsh enai Deutra. Sthn dia apnthsh ja fjname an sth jsh tou 7001 eqame 8, 15 opoiondpote fusik arijm thc morfc 7m + 1. 1.3.1 Orismc. 'Estw m nac akraioc. Do akraioi a kai b ja lgontai istimoi modulo m ( isoploipoi modulo m ) an o m diaire th diafor a b. Sthn perptwsh aut grfoume a b mod m, dhlad1 a b mod m m|a b. Gia pardeigma, qoume 7001 1 mod 7 afo o 7 diaire ton 7001 1 = 7000. Epshc 14 2 mod 8 afo o 8 diaire ton 14 (2) = 16. Epeid isqei m|a b an kai mno an m|a b, mporome na jewrsoume ston parapnw orism ti o m enai mh arnhtikc. Parathrome ti an m = 0, tte a b mod m an kai mno an 0|a b, dhlad an kai mno an a = b. 'Ara do akraioi enai istimoi modulo 0 an kai mno an enai soi. 1.3.2 Prtash. 'Estw m nac jetikc akraioc kai a, b do akraioi. Tte: 1) Isqei a b mod m an kai mno an oi a kai b afnoun to dio uploipo tan diairejon me to m. 2) Uprqei akribc nac akraioc r me a r mod m kai 0 r < m. Apdeixh. 1) Ap ton Algrijmo Diareshc qoume a = mq1 + r1, 0 r1 < m b = mq2 + r2, 0 r2 < m, 1 H nnoia thc isotimac enai exairetik qrsimh sth Jewra Arijmn all kai sthn 'Algebra pou emfanzetai pio genik up th morf twn domn phlko. Anaptqjhke de susthmatik ap ton Gauss sto rgo tou Disquitiones Arithmeticae (1801). Eke eisqjhke o sumbolismc a b mod m o opooc epikrthse ap tte. Enai axioshmewto ti, an kai qoun persei perisstera ap 200 qrnia ap thn kuklofora tou Disquitiones Arithmeti- cae, to biblo aut jewretai sgqrono kai diabzetai me eukola.
  34. 34. 34 1 Akraioi pou q1, q2, r1, r2 Z. Tte a b = m(q1 q2) + r1 r2 kai ap tic anisthtec parnoume |r1 r2| < m. Epeid m|m(q1 q2), h prohgomenh isthta dnei: m|a b an kai mno an m|r1 r2. Epeid mwc |r1 r2| < m, parnoume: m|r1 r2 an kai mno an r1 r2 = 0. Telik, m|a b an kai mno an r1 r2 = 0. 2) 'Enac r pou ikanopoie tic sunjkec thc prtashc enai to uploipo thc diareshc tou a me ton m. Gia th monadikthta parathrome ti an a r mod m, 0 r < m kai a s mod m, 0 s < m, tte ap to 1) kai th monadikthta tou upolopou diareshc me to m prokptei r = s. 1.3.3 Shmewsh. Parathrome ti isqoun oi paraktw idithtec. 1) a a mod m gia kje a Z, afo m|a a gia kje a Z. 2) an a b mod m, tte b a mod m, afo ap m|a b petai ti m| (a b), dhlad m|b a. 3) an a b mod m kai b c mod m, tte a c mod m, afo ap tic sqseic m|ab kai m|bc petai ti m|(ab)+(bc), dhlad m|ac. Deqnoume tra ti oi isotimec sumperifrontai kal se sqsh me thn prsjesh kai ton pollaplasiasm tou Z. 1.3.4 Prtash. An a b mod m kai c d mod m, tte a + c b + d mod m kai ac bd mod m Apdeixh. Ap thn upjesh qoume m|ab kai m|cd. Epomnwc, m|(a b) + (c d), dhlad m|(a + c) (b + d) pou shmanei ti a + c b + d mod m. Gia thn llh sqsh, parathrome ti ac bd = ac bc + bc bd = (a b)c + b(c d), pou enai pollaplsio tou m afo m|a b kai m|c d. 'Ara ac bd mod m. 1.3.5 Prisma. An a b mod m, tte a + c b + c mod m, ac bc mod m kai an bn mod m gia kje fusik arijm n.
  35. 35. 1.3. Isotimec 35 1.3.6 Shmewsh. Ap ta prohgomena sungoume ti mporome na qeiristo- me tic isotimec modulo m san isthtec. Mporome na pollaplasisoume na prosjsoume kat mlh do isotimec. Epshc mporome na uysoume ta mlh miac isotimac se fusik dnamh. 'Omwc qreizetai prosoq sth diaresh pwc exhgome amswc paraktw. Nmoc Diagrafc Den alhjeei genik ti ap ac bc mod m petai ti a = b mod m. Gia prdeigma qoume 7 2 4 2 mod 6, all qi 7 4 mod 6. An mwc isqei (c, m) = 1, tte ap thn isotima ac bc mod m sumperanoume ti a b mod m smfwna me to Pardeigma 1.2.8 1). Sqetik isqei to exc apotlesma. 1.3.7 Prtash. 1) An ac bc mod (cm), pou c enai diforoc tou mhdenc, tte a b mod m. 2) An ac bc mod m, pou toulqiston nac ap touc c, m enai mh mhde- nikc, tte a b mod m d , pou d = (c, m). Apdeixh. 1) An ac bc = ecm me c = 0, tte a b = em. 2) Ap thn upjesh qoume m|c(ab). 'Ara m d c d (ab). 'Omwc m d , c d = 1 kai ra m d a b. Enai faner ti isqoun ta antstrofa twn 1) kai 2) sthn prohgomenh Prtash, dhlad an a b mod m, tte ac bc mod (cm), kai an a b mod m d , pou d = (c, m), tte ac bc mod m. 1.3.8 Efarmogc. 1. Ja apodexoume ti den uprqei akraioc thc morfc 4n + 3 (n N) pou na enai to jroisma do tetragnwn akerawn. 'Estw a N thc morfc 4n+3 kai stw x, y Z me a = x2 +y2. Ap ton Algrijmo Diareshc me to 4, petai ti kje akraioc enai thc morfc 4n + r, pou r = 0, 1, 2, 3. Epeid 4n + r r mod 4, parnoume x 0 1 2 3 mod 4. Epomnwc x2 0 1 4 9 mod 4. All 4 0
  36. 36. 36 1 Akraioi mod 4 kai 9 1 mod 4. 'Ara x2 0 1 mod 4. 'Omoia qoume y2 0 1 mod 4. 'Ara qoume x2 + y2 0 1 2 mod 4, dhlad a 0 1 2 mod 4. Aut mwc enai topo giat ap thn upjesh qoume a 3 mod 4. Sthn Pargrafo 2.12 apodeiknetai na Jerhma pou qarakthrzei touc fusikoc arijmouc pou enai jroisma do tetragnwn akerawn. 2. Gia kje n N, o akraioc 33n 5n enai pollaplsioc tou 11. Epeid 33 = 27 5 mod 11, qoume 33n = (33)n = 27n 5n mod 11. 'Ara 33n 5n 0 mod 11. 3. Ja apodexoume ti den uprqoun akraioi x, y me x2 5y2 = 13. 'Estw ti uprqoun ttoioi akraioi. Tte qoume x2 5y2 13 mod 5. 'Omwc 5y2 0 mod 5 kai 13 3 mod 5. 'Ara x2 3 mod 5. Aut mwc enai topo, afo gia kje akraio x qoume x 0, 1, 2, 3 4 mod 5 kai kat sunpeia x2 0, 1, 4, 9 16 mod 5, dhlad x2 0, 1 4 mod 5. 4. Na brejon loi oi prtoi p ste oi p + 10 kai p + 14 na enai prtoi. Dokimzontac merikoc mikroc prtouc arijmoc, blpoume ti gia p = 3 oi 13 kai 17 enai prtoi. Ja dexoume tra ti den uprqei lloc p. 'Estw p prtoc me p > 3. Tte p 1 2 mod 3. An p 1 mod 3, tte p + 14 15 mod 3, dhlad p + 14 0 mod 3, kai epomnwc o p + 14 den enai prtoc afo enai pollaplsio tou 3 kai diforoc tou 3. An p 2 mod 3, tte p + 10 12 mod 3, dhlad p + 10 0 mod 3, kai kat sunpeia o p + 10 den enai prtoc. 5. Gia kje peritt n N o 1n + 2n + + (n 1)n enai pollaplsioc tou n. Parathrome ti to jroisma mpore na grafe 1n +2n + +(n1)n = k=n1 2 k=1 kn + k=n1 2 k=1 (nk)n = k=n1 2 k=1 (kn +(nk)n ). 'Eqoume nk k mod n kai ra (nk)n (k)n (1)nkn kn mod n, giat o n enai perittc. Ara kn + (n k)n 0 mod n, kai kat sunpeia 1n + 2n + + (n 1)n 0 mod n. 6. Pujagreiec tridec. Qrhsimopointac th monadikthta thc anlu- shc akerawn se ginmena prtwn all kai aplc idithtec isotimin ja prosdiorsoume louc touc akeraouc x, y, z pou qoun thn idithta x2 + y2 = z2.
  37. 37. 1.3. Isotimec 37 An h trida (x, y, z) enai mia lsh thc exswshc x2 +y2 = z2, tte kai oi (x, y, z) enai lseic kai epomnwc mporome na upojsoume ti oi x, y, z enai mh arnhtiko. Parathrome ti an d = (x, y, z) ('Askhsh 1.2.12), tte (x/d)2 + (y/d)2 = (z/d)2 kai (x/d, y/d, z/d) = 1. Epo- mnwc kje lsh thc arqikc exswshc ja enai thc morfc (dx0, dy0, dz0), pou d enai jetikc akraioc kai (x0, y0, z0) enai mia lsh me (x0, y0, z0) = 1. Ap tra kai sto exc upojtoume ti (x, y, z) = 1. Sthn perptwsh aut, oi x, y, z enai an do sqetik prtoi. Idiatera, akribc nac ap touc x, y, z enai rtioc. Parathrome ti o z den enai rtioc. Prgmati, an o z tan rtioc, tte oi x, y ja tan peritto. Aut enai topo afo ap th mia meri qoume z2 0 mod 4 kai ap thn llh z2 = x2 + y2 1 + 1 2 mod 4. Sunepc mporome na upojsoume ti: x perittc, y rtioc kai z perittc. 'Eqoume y2 = z2 x2 = (z x)(z + x). Isqurizmaste ti (zx, z+x) = 2. Prgmati, an d = (zx, z+ x), tte d|(z x) + (z + x) = 2z kai d|(z x) (z + x) = 2x, opte d|(2z, 2x) = 2(z, x) = 2. Epeid mwc oi z x, z + x enai rtioi, parnoume d = 2. Jtoume a = (z + x)/2, b = y/2, c = (z x)/2 (pou enai mh arnhtiko akraioi) opte b2 = ac. Epeid (zx, z+x) = 2, qoume (c, a) = 1, opte, smfwna me thn Efarmog 1.2.8 4), oi a, c enai tetrgwna akerawn, a = u2, c = v2. 'Ara z +x = 2u2, z x = 2v2, y2 = (2u2)(2v2) kai epomnwc x = u2 v2 , y = 2uv, z = u2 + v2 . Epiplon isqei u v, (u, v) = 1, afo (x, y, z) = 1. Apodexame ti kje lsh thc arqikc exswshc x2+y2 = z2 me x, y, z N (qwrc ton periorism (x, y, z) = 1) enai thc morfc x = d(u2 v2 )+, y = 2duv, z = d(u2 + v2 ), pou d, u, v N, u v kai (u, v) = 1. Antstrofa, me nan ekolo upologism epalhjeetai ti oi parapnw akraioi x, y, z enai lseic. Oi tridec akerawn thc morfc (d(u2 v2), 2duv, d(u2 + v2)), pou (u, v) = 1, onomzontai Pujagreiec tridec.
  38. 38. 38 1 Akraioi 7. Jerhma tou Fermat2 gia n = 4. Me th bojeia twn Pujagorewn tridwn ja apodexoume ed ti den uprqoun mh mhdeniko akraioi x, y, z pou qoun thn idithta x4 + y4 = z4. Gia ton skop aut ja apodexoume ti h exswsh x4 + y4 = z2 () den qei jetikc akraiec lseic. Aut arke giat an (a, b, c) enai lsh thc x4 + y4 = z4, tte (a, b, c2) enai lsh thc x4 + y4 = z2. 'Estw ti uprqei lsh (x, y, z) thc (*) me x, y, z > 0 jetiko akraioi. Epilgoume mia lsh me z elqisto. Tte (x, y, z) = 1 giat an p enai nac prtoc koinc diairthc twn x, y, z o p4 diaire ton x4 + y4 kai ra o p2 diaire ton z. All tte ma lsh thc (*) enai h (x/p, y/p, z/p2), prgma topo afo z/p2 < z. Grfontac (x2)2 +(y2)2 = z2, ap thn prohgomenh Efarmog parnoume x2 = u2 2 , y2 = 2u, z = u2 + 2 pou u, enai jetiko akraioi kai (u, ) = 1. Epeidh y2 = 2u qoume 4|y2 kai ra toulqiston nac ap touc u, enai rtioc. An o u enai rtioc, o enai perittc kai kat sunpeia x2 = u2 2 1 mod 4, pou enai topo (bl. Efarmog 1). Epomnwc o u enai perittc kai o rtioc, = 2 . Epeid y2 = 4u kai (u, ) = 1, oi u, enai tetrgwna akerawn: u = a2, = b2 (Pardeigma 1.1.8 4). Efarmzoume pli tic Pujagreiec tridec, aut th for sthn exswsh x2 + 2 = u2. Parathrome ti o enai rtioc kai oi x, u peritto kai ti (x, , u) = 1. Epomnwc uprqoun jetiko akraioi c, d me (c, d) = 1 ttoioi ste x = c2 d2 , = 2cd, u = c2 + d2 . Epeid b2 = = cd, sumperanoume ti oi c, d enai tetrgwna akerawn, c = e2, d = f2. 'Eqoume e4 + f4 = a2 , 2 Smfwna me to Jerhma tou Fermat , h exswsh xn + yn = zn den qei jetikc akraiec lseic tan n 3. O Fermat (per to 1637) psteue ti brke mia apdeixh, all den fhse kanna sqetik grapt rgo. Smera epikrate h poyh ti o Fermat kane ljoc. H prospjeia apdeixhc tou isqurismo tou Fermat odghse sthn anptuxh nwn shmantikn kldwn twn Majhmatikn pwc enai h Algebrik Jewra Arijmn kai h Metajetik 'Algebra. Telik o isqurismc tou Fermat apodeqthke ap ton A. Wiles (1995). H apdeixh enai exairetik dskolh kai gia thn ergasa aut apenemjh ston Wiles to Cole Prize, pou enai mia ap tic antatec diakrseic sta Majhmatik.
  39. 39. 1.3. Isotimec 39 dhlad mia lsh thc arqikc exswshc enai h (e, f, a). All qoume z = u2 + 2 = a4 + 4b4 > a4 a, dhlad z > a. Aut enai topo ap ton orism tou z. H apdeixh enai plrhc. H teqnik pou akoloujsame sthn parapnw apdeixh, smfwna me thn opoa kataskeusame mia lsh pou enai mikrterh ap mia elqisth, ofeletai ston Fermat kai onomzetai h mjodoc thc kajdou. 8. Oi kwdiko ISBN. Kje dhmosieumno biblo periqei nan kwdik, o opooc apoteletai ap 9 yhfa kai nan akraio metax 0 kai 10. O kwdikc autc onomzetai ISBN (International Standard Book Number). Ta prta 9 yhfa parqoun plhroforec gia to biblo, pwc ton tpo kai qrno kdoshc. To teleutao yhfo qrhsimeei na entopzontai ljh. An o kwdikc ISBN enai a1a2 . . . a10, pou a1, . . . , a9 {0, . . . , 9} kai a10 {0, . . . , 10} tte prpei na isqei a1 + 2a2 + + 9a9 a10 mod 11. 9. AFM. Se kje 'Ellhna forologomeno polth antistoiqe nac enniay- fioc Arijmc Forologiko Mhtrou (AFM). 'Opwc kai sto prohgomeno pardeigma, to teleutao yhfo uprqei gia na apofegontai ljh kai na entopzontai plasto AFM. Sugkekrimna, an a1 . . . a9 enai o AFM tte prpei na isqei 28 a1 + 27 a2 + + 2a8 a9 mod 11, tan to arister mloc den enai isodnamo me to 10 mod 11. Diafore- tik, prpei na isqei a9 = 0. Askseic 1.3 1) Apodexte ti an a b mod m kai n|m tte a b mod n. 2) Exetste an isqoun ta paraktw isotimec i) 2004 1003 mod 11 ii) (7 a)2 a2 mod 7 gia kje a Z iii) (1 2n)2 (4n + 1)10 mod 4n gia kje n Z iv) (6n + 5)2 1 mod 4 gia kje n Z 3) Apodexte ti
  40. 40. 40 1 Akraioi i) a b mod 2 a2 b2 mod 4 ii) a b mod 3 a3 b3 mod 9 4) Apodexte ti a b mod m an kai mno an a2 + b2 2ab mod m2. Updeixh: Efarmog 1.2.8 7). 5) Apodexte ti gia kje peritt akraio a isqei a2 1 mod 8. 6) Apodexte ti gia kje a Z isqei a2 0, 1 4 mod 8. Kat sunpeia o arijmc 200340067085 den enai tetrgwno akeraou. 7) Apodexte ti kannac akraioc thc morfc 3m +3n +1, pou m, n jetiko akraioi, den enai tetrgwno akeraou. Updeixh: Ergastete mod 8. 8) Apodexte ti den uprqei akraioc thc morfc 4n + 2, pou n Z, pou enai diafor do tetragnwn akerawn. 9) Gia kje n N apodexte ti 4n 1 + 3n mod 9. 10) Na brejon loi oi akraioi 0 x 101 pou ikanopoion x2 1 mod 101. 11) 'Estw p prtoc arijmc. Apodexte ti an gia ton akraio x isqei x2 x mod p, tte x 0 1 mod p. 12) Poi enai to uploipo thc diareshc tou 100100 me to 11; 13) Apodexete ti kje hmerologiak toc (dsekto mh) qei ma toulqi- ston Trth kai 13. Updeixh: Metrste modulo 7 arqzontac ap thn 13h Ianouarou. 14) Apodexte ti (a, m) = (b, m) an a b mod m. Exetste an alhjeei to antstrofo. 15) 'Enac akraioc arijmc (se dekadik graf) diairetai me to 9 an kai mno an to jroisma twn yhfwn tou, i ai, diairetai me to 9. 16) 'Enac akraioc arijmc ak a1a0 (se dekadik graf) diairetai me to 11 an kai mno an o arijmc i (1)iai diairetai me to 11. 17) 'Enac akraioc arijmc ak a1a0 (se dekadik graf) diairetai me to 5 an kai mno an o arijmc a0 diairetai me to 5.
  41. 41. 1.3. Isotimec 41 18) i) 'Estw a b mod m kai a b mod n. Apodexte ti a b mod e, pou e = (m, n). ii) Na brejon loi oi akraioi x pou ikanopoion x 7 mod 8 kai x 7 mod 9 19) Na brejon lec oi tridec (p, p + 4, p + 8) pou oi p, p + 4, p + 8 enai prtoi arijmo. 20) Apodexte ti den uprqoun akraioi x, y me 7x2 15y2 = 1. 21) Apodexte ti den uprqoun mh mhdeniko akraioi x, y, z me x2 +y2 = 3z2. Updeixh:Mjodoc thc kajdou: 'Estw (x, y, z) Z Z Z mia lsh me x jetik kai elqisto. Ap x2 +y2 = 3z2, sumpernate ti kajna ap ta x, y, z enai pollaplsio tou 3. Aplopointac lambnoume mia na lsh (x0, y0, z0) me 0 < x0 < x. Aut enai topo. 22) An o x2 + y2 + z2 enai pollaplsio tou 5 (pou x, y, z Z) apodexte ti na toulqiston ap touc x, y, z enai pollaplsio tou 5. 23) Apodexte ti an o m N enai tetrgwno akeraou kai kboc akeraou, tte enai thc morfc 7k 7k + 1. 24) Apodexte ti gia kje n N isqei 11|33n+1 + 24n+3. 25) Apodexte ti gia kje n N isqei 21|4n+2 + 52n+1. 26) Apodexte ti gia kje n N isqei i) 10n + 3 4n+2 4 mod 9 ii) (n + 1)2n + 4n2n+1 den enai pollaplsio tou 3. 27) 'Estw m, n N me ton n peritt. Tte o akraioc in + (m i)n enai pollaplsioc tou m gia kje i = 0, . . . , m. 28) Apodexte ti an oi m, n enai peritto tte 1m + 2m + + (n 1)m 0 mod n. 29) 'Estw n perittc fusikc arijmc. Apodexte ti kje akraioc enai is- timoc modulo n me akribc nan akraio ap touc n 1 2 , n 3 2 , . . . , 1, 0, 1, . . . , n 3 2 , n 1 2 .
  42. 42. 42 1 Akraioi 1.4 Oi Akraioi modulo m Sthn prohgomenh Pargrafo diapistsame ti oi isotimec modulo m ikano- poion arijmhtikc idithtec parmoiec me idithtec twn akerawn. Sthn Par- grafo aut ja kataskeusoume na peperasmno snolo Zm kai ja orsoume kat fusik trpo to jroisma kai to ginmeno do stoiqewn tou. Ja dome ti oi arijmhtik sto Zm qei mesh sqsh me thn arijmhtik isotimin modulo m. To snolo Zm me tic prxeic autc parousizei idiatero endiafron, giat afenc men oi upologismo se aut epitrpoun suqn thn exagwg me sntomo kai komy trpo qrsimwn sumperasmtwn pou aforon akeraouc, afetrou de aut apotele na shmantik pardeigma do genikterwn ennoin pou ja meletsoume stic epmenec Enthtec. Sqseic isodunamac Ja qreiastome ed (all kai se epmenec Enthtec) basik stoiqea ap tic isodunamec ta opoa ja upenjumsoume. Mia sqsh se na snolo A enai na uposnolo tou kartesiano ginomnou A A = {(a, b)|a, b A}. 'Estw X mia sqsh sto A. Ant na grfoume (a, b) X, suqn qrhsimo- poiome ton sumbolism a X b. Epshc, pollc forc ja grfoume a b, tan enai faner poio snolo X ennoome. Sunepc oi sumbolismo (a, b) X kai a b enai isodnamoi. Mia sqsh sto mh ken snolo A ja lgetai sqsh isodunamac sto A an isqoun oi paraktw idithtec. 1) a a gia kje a A (anaklastik idithta) 2) an a b, tte b a (summetrik idithta), kai 3) an a b kai b c, tte a c (metabatik idithta). 1.4.1 Paradegmata. 1) 'Estw A na mh ken snolo. Jewrome th sqsh pou orzetai wc exc: a b an kai mno an a = b. Tte enai safc ti orzetai mia sqsh isodunamac sto A. 2) 'Estw A to snolo twn shmewn tou pragmatiko epipdou. Jewrome th sqsh pou orzetai wc exc: P Q an kai mno an ta shmea P, Q isapqoun ap thn arq twn axnwn. Tte orzetai mia sqsh isodunamac sto A.
  43. 43. 1.4. Oi Akraioi modulo m 43 3) 'Estw A = R. Jewrome th sqsh sto R pou orzetai wc exc: a b an kai mno an a b Z. Ekola diapistnetai ti orzetai mia sqsh isodunamac sto R. 'Estw X mia sqsh isodunamac sto snolo A. An a, b A me a b, ja lme ti to a enai isodnamo me to b ( ti ta a kai b enai isodnama, prgma pou mporome na pome lgw thc summetrikc idithtac). To snolo [a] = {x A|x a}, dhlad to snolo twn stoiqewn tou A pou enai isodnama me to a, onomzetai klsh isodunamac tou a. Profanc qoume a [a] afo a a. 1.4.2 Paradegmata. (sunqeia) H arjmhsh ed anafretai sta prohgomena paradegmata. 1) Gia kje a A, isqei [a] = {a}, 2) Gia kje shmeo P, h klsh isodunamac [P] enai to snolo twn shmewn tou kklou pou dirqetai ap to P kai qei kntro thn arq twn axnwn. 3) Gia kje a R, isqei [a] = {a + m R|m Z}. Eidik qoume [k] = Z gia kje k Z. H epmenh Prtash perigrfei tic idithtec twn klsewn isodunamac pou ja qreiastome. 1.4.3 Prtash. 'Estw X mia sqsh isodunamac sto snolo A kai a, b A. Tte 1. [a] = [b] an kai mno an ta a, b enai isodnama. 2. [a] [b] = an kai mno ta a, b den enai isodnama. 3. To snolo A mpore na parastaje wc xnh nwsh klsewn isodunamac. Apdeixh. 1. 'Estw [a] = [b]. Tte a [a] = [b], opte a b, ap ton ori- sm thc klshc isodunamac. Antstrofa, stw a b kai x A. An x [a], tte x a. Epeid a b, h metabatik idithta dnei x b, opte qoume x [b]. Sunepc [a] [b]. Me parmoio trpo apodeiknetai ti [b] [a]. 'Ara [a] = [b]. 2. 'Estw ti ta a kai b den enai isodnama kai stw x [a][b]. Ap th sqsh x [a] sumperanoume ti a x. 'Omoia, qoume ti x . Epomnwc qoume a b, pou enai topo. Tloc an ta a, b enai isodnama, tte [a] = [b], pwc
  44. 44. 44 1 Akraioi edame prohgoumnwc, opte [a] [b] = [a] = [b] = . 3. Enai faner ti A = aA [a]. H nwsh aut den enai anagkastik xnh. 'E- stw ti to snolo twn diakekrimnwn klsewn isodunamac enai to {[b]|b B} gia kpoio B A. Tte qoume A = bB [b] kai epiplon ap to 2 sumpera- noume ti [b] [b ] = , gia kje b, b B me b = b . 'Estw X mia sqsh isodunamac sto snolo A kai a A. Kje stoiqeo thc klshc isodunamac [a] onomzetai antiprswpoc thc klshc [a]. Ap thn prohgomenh Prtash petai ti na b A enai antiprswpoc thc klshc [a], an kai mno an [a] = [b], dhlad an kai mno an a b. 'Estw ti to snolo twn diakekrimnwn klsewn isodunamac enai to {[b]|b B} gia kpoio B A. Kje ttoio snolo B onomzetai na plrec ssthma antiprospwn thc sqshc isodunamac X. To snolo Zm Ja efarmsoume tra ta parapnw se ma endiafrousa perptwsh. 'Estw m nac fusikc arijmc. Sto snolo Z jewrome th sqsh pou orzetai wc exc a b a b mod m (1) Sth Shmewsh 1.3.3 edame ti h parapnw sqsh enai mia sqsh isodunamac. H klsh isodunamac tou a Z enai [a] = {x Z|x a mod m} = {x Z|m diaire ton x a} = {x Z|x a = km, gia kpoio k Z} = {a + km Z|k Z}. (2) Dhlad, ta stoiqea tou sunlou [a] enai thc morfc a + km, k Z. Gia ton lgo aut sunhjzetai o sumbolismc [a] = a + mZ. 'Otan jloume na dhlsoume thn exrthsh tou sunlou [a] ap to m, sunjwc qrhsimopoiome ton sumbolism [a]m, a mod m. Ap thn Prtash 1.4.3 qoume ti [a] = [b] a b mod m. (3) To snolo twn klsewn isodunamac pou orzontai ap thn sqsh isodunamac (1) sumbolzetai me Zm, Zm = {[a] | a Z}. Gia pardeigma, stw m = 2. Tte ap th (2) parnoume [0] = {x Z|x rtioc} kai [1] = {x Z|x perittc}. Epeid oi akraioi . . . , 4, 2, 0, 2, 4,
  45. 45. 1.4. Oi Akraioi modulo m 45 . . . enai istimoi mod 2 qoume, lgw thc (3), ti . . . = [4] = [2] = [0] = [2] = [4] = . . .. Epshc, afo oi akraioi . . . , 3, 1, 1, 3, . . . enai istimoi mod 2, qoume . . . = [3] = [1] = [1] = [3] = . . .. 'Ara Z2 = {[0], [1]}. Parathrome ti Z2 = {[4], [3]} = {[4], [1]} = . . . kai genik Z2 = {[a0], [a1]}, pou ai i mod 2. 'Estw tra m = 3. Tte ap th (2) parnoume [0] = {x Z|x = 3k, k Z}, [1] = {x Z|x = 3k + 1, k Z} kai [2] = {x Z|x = 3k + 2, k Z}. Dhlad, [0] = {. . . , 3, 0, 3, ...}, [1] = {. . . , 2, 1, 4, . . .} kai [2] = {. . . , 1, 2, 5, . . .}. Ap th sqsh (3) qoume ti . . . = [3] = [0] = [3] = . . . pwc epshc . . . = [2] = [1] = [4] = . . . kai . . . = [1] = [2] = [5] = . . .. 'Ara Z3 = {[0], [1], [2]}. Parathrome ti qoume Z3 = {[3], [2], [1]} = {[3], [1], [2]} = . . . kai genik Z3 = {[a0], [a1], [a2]} pou ai i mod 3. 1.4.4 Prtash. Gia kje jetik akraio m qoume Zm = {[0], [1], . . . , [m1]}. Pio genik qoume Zm = {[a0], [a1], . . . , [am1]}, pou oi ai enai akraioi ttoioi ste ai i mod m gia kje i. Apdeixh. Parathrome ti oi klseic isodunamac [ai], i = 0, . . . , m 1 enai diakekrimnec. Prgmati, an [ai] = [aj] me 0 i, j m 1, tte ap thn (3) qoume i j mod m, opte ap thn Prtash 1.3.2 2) qoume i = j. Kje klsh isodunamac [a] ankei sto snolo {[a0], [a1], . . . , [am1]}. Prgmati, ap ton algrijmo diareshc qoume = qm + r, pou 0 r m 1, opte a r mod m kai ra [a] = [r] = [ar]. Sunepc apodexame ti Zm {[a0], [a1], . . . , [am1]}. Epomnwc Zm = {[a0], [a1], . . . , [am1]}. To Zm onomzetai to snolo twn akerawn modulo m kai ta stoiqea tou onomzontai klseic isotimac modulo m, klseic upolopwn modulo m. Parathrseic 1) 'Enac diaisjhtikc trpoc na skeftmaste to snolo Zm enai o exc. Grw ap nan kklo me mkoc perifreiac m tulgoume ton xona twn pragmatikn arijmn. Tte ta shmea . . . , m, 0, m, 2m, . . . tou xona ja tautiston pnw ston kklo. Omowc ja tautiston ta shmea . . . , m+1, 1, m+1, 2m+1, . . ..
  46. 46. 46 1 Akraioi 2) Sthn prohgomenh Prtash eqame upojsei ti o m enai jetikc. An m = 0, tte ap th sqsh (1) qoume a b an kai mno an a = b. Sunepc h klsh isodunamac tou a apoteletai ap na mno stoiqeo, [a] = {a}. Tte, tautzontac to snolo {a} me to stoiqeo a, mporome na jewrsoume ti to snolo twn klsewn upolopwn modulo 0 enai to Z. 3) An m = 1, tte ap th sqsh (1) qoume a b gia kje do akeraouc a, b. Sunepc h klsh isodunamac enc akeraou a enai lo to snolo Z kai ra to snolo twn klsewn upolopwn modulo 1 enai na monosnolo. H prsjesh kai o pollaplasiasmc sto Zm Kataskeusame to snolo Zm me th bojeia miac sqshc isodunamac sto snolo Z. Sto Z, mwc, gnwrzoume pwc na prosjtoume kai na pollaplasi- zoume stoiqea. Sunepc enai elogo to erthma an mporome na prosjtoume kai na pollaplasizoume stoiqea tou Zm me anlogo trpo. H prsjesh (antstoiqa, o pollaplasiasmc) akerawn antistoiqe se kje zegoc (a, b) akerawn monadik akraio, ton a+b (antstoiqa, ton ab). Dhlad qoume apeikonseic + : Z Z Z, (a, b) a + b : Z Z Z, (a, b) ab. Me th bojeia autn, orzoume tic antistoiqec, + : Zm Zm Zm, ([a], [b]) [a + b] : Zm Zm Zm, ([a], [b]) [ab]. Apodeiknoume tra ti oi antistoiqec autc enai apeikonseic. Prpei na dexoume ti: an ([a], [b]) = ([c], [d]), dhlad an [a] = [c] kai [b] = [d], tte petai ti [a + b] = [c + d] kai [ab] = [cd]. Me lla lgia, an a c mod m kai b d mod m, tte a + b c + d mod m kai ab cd mod m. 'Omwc aut isqei ap thn Prtash 1.3.4. Gia pardeigma, sto Z3 qoume [4] + [2] = [6] = [0], [2][2] = [4] = [1]. Sto Z10 qoume [4] + [7] = [11] = [1], [4][7] = [28] = [8], [5][6] = [30] = [0]. Blpoume ti h prsjesh kai o pollaplasiasmc klsewn upolopwn angontai sth prsjesh kai ston pollaplasiasm antiprospwn touc, dhlad akerawn. Gia thn prsjesh kai ton pollaplasiasm pou orsame sto Zm isqoun oi paraktw idithtec pou jumzoun genikc idithtec thc Arijmhtikc. Gia kje [a], [b], [c] Zm qoume ti: 1. ([a] + [b]) + [c] = [a] + ([b] + [c])
  47. 47. 1.4. Oi Akraioi modulo m 47 2. [a] + [0] = [0] + [a] 3. [a] + [a] = [a] + [a] = [0] 4. [a] + [b] = [b] + [a] 5. ([a][b])[c] = [a]([b][c]) 6. [a]([b] + [c]) = [a][b] + [a][c] 7. ([a] + [b])[c] = [a][c] + [b][c] 8. [a][b] = [b][a] 9. [a][1] = [1][a] = [a] Oi apodexeic enai aplc kai paralepontai. Oi idithtec 1-9 ja efarmzontai sta paraktw qwrc idiaterh mnea. Paratrhsh Prpei na toniste ed, ti an kai h prsjesh kai o pollapla- siasmc stoiqewn tou Zm qoun idithtec pou jumzoun thn prsjesh kai ton pollaplasiasm akerawn, uprqoun shmantikc diaforc. Gia pardeigma, sto Z to ginmeno do mh mhdenikn stoiqewn enai mh mhdenik. Sto Z6, mwc, qoume [2][3] = [0]. Epshc, an oi akraioi a, b, c enai ttoioi ste ac = bc kai c = 0, tte a = b. Sto Z6, mwc, qoume [1][3] = [5][3] me [3] = [0] kai [1] = [5]. Antistryima stoiqea sto Zm Sth melth thc Arijmhtikc tou Zm (dhlad twn idiottwn thc prsjeshc kai tou pollaplasiasmo tou Zm) enai fusik na jewrsoume poluwnumikc exisseic sto Zm, dhlad poluwnumikc exisseic me suntelestc stoiqea tou Zm. Ma ap tic aplosterec ap autc tic exisseic enai h [a][x] = [b]. Sthn perptwsh pou uprqei klsh [a ] Zm me thn idithta [a ][a] = [1] mporome ekola na lsoume thn exswsh pollaplasizontc thn me thn [a ], [a][x] = [b] [a ]([a][x]) = [a ][b] [x] = [a b]. 'Omwc den uprqei pnta ttoia klsh [a ]. Gia pardeigma, stw [2] Z6. An uprqe [a ] Z6 me [2][a ] = [1], tte [2a ] = [1] kai ra 2a 1 mod 6, dhlad 2a = 1 + 6n, n Z, pou enai topo. Odhgomaste tsi ston epmeno orism. 'Ena stoiqeo [a] Zm lgetai antistryimo an uprqei [a ] Zm me thn idithta [a][a ] = [1], dhlad an uprqei akraioc a ttoioc ste aa 1 mod m. Sthn perptwsh aut, to stoiqeo [a ] onomzetai antstrofo tou [a]. Epshc ja lme ti na antstrofo modulo m tou akeraou a enai o akraioc
  48. 48. 48 1 Akraioi a . Gia pardeigma, sto Z6 to[5] enai antistryimo afo [5][5] = [1], en to [2] den enai, pwc edame prin. Mlista, ta mna antistryima stoiqea tou Z6 enai ta [1], [5]. Sto Z7 to[2] enai antistryimo afo [2][4] = [1]. To snolo twn antistryimwn stoiqewn tou Zm sumbolzetai me U(Zm). 'Estw [a] na antistryimo stoiqeo tou Zm. Tte uprqei monadik stoiqeo [a ] Zm me thn idithta [a][a ] = [1]. Prgmati, an eqame [a][a ] = [1] kai [a][a ] = [1], tte [a ] = [a ][1] = [a ]([a][a ]) = ([a ][a])[a ] = ([a ][a])[a ] = [1][a ] = [a ]. Sthn epmenh Prtash prosdiorzonta ta antistryima stoiqea tou Zm. 1.4.5 Prtash. To stoiqeo [a] Zm enai antistryimo an kai mno an (a, m) = 1. Apdeixh. 'Estw ti [a][b] = 1. Tte ab 1 mod m, dhlad ab = mn + 1 gia kpoio n Z. Ap thn teleutaa sqsh prokptei ti (a, m) = 1. Antstrofa, stw (a, m) = 1. Tte uprqoun x, y Z ttoioi ste ax + my = 1 (Jerhma 1.2.4). 'Eqoume [ax + my] = [1] [ax] + [my] = [1] [a][x] + [m][y] = [1] [a][x] + [0][y] = [1] [a][x] = [1], dhlad to [a] enai antistryimo. Shmewsh H parapnw apdeixh periqei nan praktik trpo upologismo tou antistrfou (efson aut uprqei)tou [a]. Me to sumbolism thc ap- deixhc, to antstrofo tou [a] enai to [x]. 'Enac ttoioc akraioc x mpore na breje qrhsimopointac ton Eukledeio algrijmo, pwc gnwrzoume ap thn Pargrafo 1.2. Ja dome amswc paraktw na sqetik pardeigma. 1.4.6 Efarmog. Ja brejon loi oi akraioi x ttoioi ste 8x 11 mod 15. Ergazmenoi sto Z15 qoume [8x] = [11], dhlad [8][x] = [11]. (1) Epeid (8, 15) = 1, to stoiqeo [8] enai antistryimo sto Z15 smfwna me thn prohgomenh Prtash. 'Estw [8][y] = [1]. Pollaplasizontac thn (1) me [y] parnoume [x] = [11y], kai sunepc arke na prosdiorsoume nan akraio y. Qrhsimopointac ton Eukledeio algrijmo gia to zegoc (8, 15) qoume 15 = 1 8 + 7, 8 = 1 7 + 1.
  49. 49. 1.4. Oi Akraioi modulo m 49 'Ara 1 = 8 1 7 = 8 1 (15 1 8) = 8 2 + 15(1). Sunepc [1] = [8 2] + [15 (1)] = [8][2] kai ra mporome na jsoume y = 2. Telik [x] = [11 2] = [22] = [7], dhlad x = 15n + 7, n Z. To snolo twn antistryimwn stoiqewn tou Zm qei endiafrousec idith- tec wc proc ton pollaplasiasm tou Zm. Gia pardeigma, isqei [a]k = [1] gia kje [a] U(Zm), pou k enai to pljoc twn stoiqewn tou U(Zm). Aut ja apodeiqje sthn Pargrafo 1.6. Ed ja apodexoume thn eidik perptwsh pou o m = p enai prtoc. Sthn perptwsh aut, ap thn Prtash 1.4.4 kai thn Prtash 1.4.5 qoume ti k = p 1. 1.4.7 Jerhma (Mikr Jerhma tou Fermat). 'Estw a Z kai p nac prtoc arijmc. Tte ap a mod p. An epiplon o p den diaire ton a, tte ap1 1 mod p. Apdeixh. Gia na dexoume thn prth sqsh, jewrome arqik thn per- ptwsh a N kai qrhsimopoiome epagwg ston a. Gia a = 0, h sqsh enai profanc. Upojtontac ti isqei h isotima gia ton a, ja thn apodexoume gia ton a + 1. Ap to diwnumik anptugma qoume (a + 1)p = ap + p 1 ap1 + p 2 ap2 + + p p 1 a + 1. Isqurizmaste ti oi akraioi p i = (p i + 1) (p 1)p 1.2 . . . i enai pollaplsioi tou p tan i = 1, 2, . . . , p 1. Prgmati, grfontac (p i + 1) . . . (p 1)p = p i 1 2 i parathrome ti o p diaire to arister mloc kai ra to dexi. Afo o p enai prtoc ja diaire nan toulqiston pargonta lgw thc Paratrhshc 1.2.6 1. 'Omwc o p den diaire kanna ap touc 1, 2, . . . i, opte diaire ton p i . Epomnwc qoume ti (a + 1)p ap + 1 mod p. Ap thn epagwgik upjesh isqei ap a mod p kai sunepc (a+1)p a+1 mod p, pou enai to zhtomeno.
  50. 50. 50 1 Akraioi 'Eqoume apodexei thn prth sqsh gia a N. 'Estw tra a Z, a < 0. An o p enai perittc qoume ap = (a)p (a) mod p ap thn perptwsh thc isotimac pou apodexame prin. 'Ara ap a mod p. 'An p = 2, tte a2 = (a)2 a mod 2. All a a mod 2 kai kat sunpeia a2 a mod 2. Ja apodexoume tra th deterh sqsh. Epeid qoume (p, a) = 1, to stoiqeo [a] enai antistryimo sto Zp (Prtash 1.4.5). 'Estw [b][a] = [1]. Pollaplasizontac th sqsh [ap] = [a] me [b] prokptei ti [ap1] = [1]. 1.4.8 Paratrhsh. Ap th deterh isotima sto Mikr Jerhma tou Fermat qoume ti an o p enai prtoc kai o a Z den diairetai me ton p, tte to antstrofo tou [a] sto Zp enai to [ap2]. 1.4.9 Paradegmata. 1) To Mikr Jerhma tou Fermat mac dieukolnei na upologsoume up- loipa diairsewn meglwn arijmn me prtouc. Gia pardeigma, ac upo- logsoume to uploipo thc diareshc tou 222555 me to 7. Epeid qoume 222 = 31 7 + 5 parnoume 222 5 mod 7. 'Ara 222555 5555 mod 7. Ap th sqsh 555 = 926+3 parnoume 5555 = (56)92 53. Ap to Mikr Jerhma tou Fermat qoume 56 1 mod 7. Sunepc 5555 53 mod 7. All 53 (2)3 8 6 mod 7. To zhtomeno uploipo enai 6. 2) 'Estw p nac prtoc arijmc kai a, b Z me ap bp mod p. Tte ap bp mod p2 Prgmati, qrhsimoipointac to Mikr Jerhma tou Fermat, ap thn upjesh sumperanoume ti a b mod p. Sunepc b = a + kp, k Z. Tte qrhsimopointac to diwnumik anptugma qoume bp ap = (a + kp)p ap = p 1 ap1 (kp) + p 2 ap2 (kp)2 + + p p (kp)p . Enai profanc ti sto dexi mloc kje prosjetoc diairetai me ton p2.
  51. 51. 1.4. Oi Akraioi modulo m 51 3) Gia kje n N isqei 42|n7 n. 'Eqoume 42 = 2 3 7. Ap to Pardeigma 1.2.8 2), arke na apodeiqje ti isqei kje mia ap tic isotimec n7 n mod 7 n7 n mod 3 n7 n mod 2 H prth isqei ap to Mikr Jerhma tou Fermat gia p = 7. Gia th deterh parathrome ti efarmzontac do forc to Mikr Jerhma tou Fermat gia p = 3 qoume n7 = (n3 )2 n n2 n n mod 3 Me parmoio trpo ( kai mesa) apodeiknetai kai h trth isotima. Askseic 1.4 1) Poia enai ta antistryima stoiqea tou Z10; Gia kajna ap aut u- pologste to antstrofo stoiqeo. Poia ap ta antistryima stoiqea sumpptoun me to antstrofo touc; 2) Na brejon loi oi akraioi x ttoioi ste 12x 11 mod 13. 3) Na luje sto Z127 h exswsh [58][x] = [3]. 4) Alhjeei ti h exswsh [4][x] = [3] qei lsh sto Z6; 5) Apodexte ti h exswsh [a][x] = [b] qei lsh sto Zm an kai mno an (a, m)|b. 6) Na lujon oi paraktw exisseic [x]2 = [1] sto Z8 [x]4 = [1] sto Z5 [x]3 = [1] sto Z5 [x]2 + [3][x] + [2] = [0] sto Z6 [x] + [x] + [x] = [0] sto Z3.
  52. 52. 52 1 Akraioi 7) Apodexte ti o akraioc p enai prtoc an kai mno an to pljoc twn antistryimwn stoiqewn tou Zp enai p 1. 8) Apodexte ti gia kje n N isqei n5 n mod 30. 9) Apodexte ti gia kje n N isqei n49 n mod 1547. Updeixh: 1547 = 7 13 17. 10) Apodexte ti gia kje n N isqei (n + 1)9 + 4n5 1 mod 5 11) Apodexte ti n12 + 12n 5 mod 11 an kai mno an n 2, 9 mod 11. 12) Poio enai to uploipo thc diareshc tou 100100 me to 13; 13) Poia hmra thc ebdomdac ja enai 333444 hmrec ap smera; 14) Na breje na stoiqeo [a] tou U(Z5) ste kje llo stoiqeo tou U(Z5) na enai thc morfc [a]n, n N. Uprqei ttoio stoiqeo sto U(Z8); 15) 'Estw p prtoc kai a Z pou den enai pollaplsio tou p. Apodexte ti o elqistoc jetikc akraioc n gia ton opoo isqei sto U(Zp) ti [a]n = [1] enai diairthc tou p 1. 16) Upologste to jroisma lwn twn stoiqewn tou Zm tan m = 3, 4, 5, 6. Apodexte ti an o m enai perittc, tte to jroisma lwn twn stoiqewn tou Zm enai so me [0]. Me t isotai to en lgw jroisma tan o m enai rtioc; 17) 'Estw p prtoc arijmc me p 3 mod 4. Apodexte ti den uprqei [a] Zp me [a]2 = [1]. 18) 'Estw Zm = {[a1], . . . [am]} kai [a] Zm. Apodexte ti ta stoiqea [a] + [ai], pou i = 1, . . . , m, enai diakekrimna kai epomnwc Zm = {[a + a1], . . . , [a + am]}. 'Estw epiplon ti [a] = [0]. Alhjeei ti ta stoiqea [a][ai], i = 1, . . . , m, enai diakekrimna; 19) Exetste an alhjeei ti to jroisma do antistryimwn stoiqewn tou Zm enai antistryimo.
  53. 53. 1.5. Diofantikc Exisseic kai Isotimec 53 1.5 Diofantikc Exisseic kai Isotimec Poll majhmatik problmata pou sunantme sthn kajhmerin mac zw an- gontai se exisseic stic opoec epizhtome lseic pou na enai akraioi arijmo. Oi exisseic autc thc morfc pazoun shmantik rlo sth Jewra Arijmn kai sthn 'Algebra. Mia exswsh thc morfc f(x1, . . . xn) = 0, pou to f(x1, . . . xn) enai na polunumo twn x1, . . . xn me akeraouc suntelestc, onomzetai Diofantik tan mac endiafroun mno oi akraiec lseic thc. Ja asqolhjome ed me thn prth mh tetrimmnh Diofantik exswsh, ax + by = c, kai ja perigryoume plrwc tic lseic thc. Me th qrsh autc ja lsoume thn isotima ax b mod m. Tloc ja asqolhjome me sustmata isotimin (Kinezik Jerhma Upolopwn). 1.5.1 Jerhma. 'Estw a, b, c Z ttoioi ste toulqiston nac ap touc a, b den enai mhden. Jtoume d = (a, b). 1) An o d den diaire ton c, tte h Diofantik exswsh ax + by = c den qei lseic. 2) An o d diaire ton c, tte h Diofantik exswsh qei peirec lseic. Epi- plon an (x0, y0) enai ma lsh, tte kje llh lsh qei th morf x = x0 + b d n, y = y0 a d n, n Z. (1) Apdeixh. 1) Upojtoume ti o d den diaire ton c. 'Estw ti uprqoun akraioi x, y ttoioi ste ax + by = c. Afo d|a kai d|b, parnoume d|c, pou enai topo. 2) Upojtoume ti o d diaire ton c. Epeid (a, b) = d, uprqoun s, t Z me d = as + bt (2) Afo d|c, qoume c = de, pou e Z. Ap thn (2) parnoume c = de = a(se) + b(te), pou shmanei ti ma lsh enai h x0 = se, y0 = te. Ja dexoume ti h dojesa Diofantik exswsh qei peirec lseic. 'Estw x = x0 + b d n kai y = y0 a d n, n Z. Ekola epalhjeetai me prxeic ti ax + by = c.
  54. 54. 54 1 Akraioi Tra ja dexoume ti kje lsh enai thc morfc (1). 'Estw x0, y0, x, y Z me ax + by = c kai ax0 + by0 = c. Afairntac kat mlh parnoume a(x x0) = b(y0 y) (3) kai ra a d (x x0) = b d (y y0). Afo a d , b d = 1, qoume ti a d y y0. 'Ara uprqei n Z me y0 y = a d n, dhlad y = y0 a d n . An o a den enai mhdn, tte ap thn (3) parnoume x = x0 + b d n. H apdeixh sthn perptwsh pou o b den enai mhdn enai parmoia. 1.5.2 Paradegmata. 1) H Diofantik exswsh 4x + 8y = 10 den qei lseic afo (4, 8) = 4 pou den diaire ton 10. 2) H Diofantik exswsh 21x+14y =70 qei peirec lseic afo (21, 14) = 7 pou diaire to 70. Brskoume tic lseic wc exc: Ap ton Eukledeio algrijmo qoume 21 = 1 14 + 7, 14 = 2 7 = 0. Epomnwc 7 = 1 21 + (1) 14 kai ra 70 = 10 21 + (10) 14. 'Etsi ma lsh enai h x0 = 10, y0 = 10. Epomnwc kje lsh, smfwna me to Jerhma 1.5.1, enai thc morfc x = 10 + 2n kai y = 10 3n, pou n Z. 3) Jloume na agorsoume grammatshma twn 0.4 Eur kai 0.6 Eur gia mia epistol pou kostzei 8 Eur. Poioc enai o elqistoc arijmc gram- matosmwn 0.4 Eur pou apaitontai; Ja lsoume th Diofantik exswsh 4x + 6y = 80 kai ja brome th lsh (x, y) pou o x enai mh arnhtikc akraioc kai elqistoc kai y 0. Afo (4, 6) = 2 pou diaire to 80, uprqoun lseic. Ap ton Eukledeio Al- grijmo parnoume 80 = (40) 4 + 40 6 pou shmanei ti ma lsh enai h x0 = 40, y0 = 40. 'Ara kje lsh enai thc morfc x = 40 + 3n, y = 40 2n. Epeid x 0 qoume n 14. 'Ara h zhtomenh lsh prokptei gia n = 14 kai enai h x = 2, y = 12. To prohgomeno Jerhma mac bohj na meletsoume isotimec. 1.5.3 Jerhma. 'Estw a, b, m Z me m = 0 kai d = (a, m). Gia tic lseic thc isotimac ax b mod m isqoun ta exc: 1) an d b, tte den uprqoun lseic, kai
  55. 55. 1.5. Diofantikc Exisseic kai Isotimec 55 2) an d|b tte uprqoun akribc d mh isodnamec lseic modulo m. Apdeixh. H isotima ax b mod m enai isodnamh me th Diofantik exswsh ax my = b. Oi lseic autc perigrfontai ap to Jerhma 1.5.1: an d b tte den uprqoun lseic, en an d|b tte uprqoun peirec lseic pou dnontai ap tic sqseic x = x0 + m d t, y = y0 a d t (t Z), pou (x0, y0) enai mia sugkekrimnh lsh thc exswshc. 'Estw t1, t2 Z kai x1 = x0 m d t1 kai x2 = x0 m d t2, Ja apodexoume tra ti x1 x2 mod m t1 t2 mod d. Prgmati, an x1 x2 mod m, tte m d t1 m d t2 mod m. Ap thn Prtash 1.3.7 prokptei ti t1 t2 mod d. Antstrofa, an t1 t2 mod d, tte m d t1 m d t2 mod m kai ra x1 x2 mod m. 'Ara apodexame ti uprqoun akribc d mh isodnamec lseic modulo m. Ma epilog d mh isodunmwn lsewn modulo m dnetai ap th sqsh x = x0 m d t, (4) gia t = 0, 1, . . . , d 1. Shmeiseic 1) An c Z enai mia lsh thc isotimac ax b mod m tte kje c Z me c c mod m ja enai lsh thc isotimac lgw thc Prtashc 1.3.4. Sunepc, ap tra kai sto exc tan lme lsh miac isotimac ax b mod m ennoome mia klsh upolopwn modulo m, ttoia ste kje stoiqeo x thc klshc ikanopoie thn isotima. 'Etsi sth deterh perptwsh tou prohgomenou Jewrmatoc, ja lme ti h isotima ax b mod m qei akribc d lseic. 2) Epeid h isotima ax b mod m enai isodnamh me thn exswsh [a][x] = [b] sto Zm, blpoume ti mia prwtobjmia exswsh sto Zm mpore na mhn qei lseic, na qei akribc mia lsh na qei perissterec lseic.
  56. 56. 56 1 Akraioi 1.5.4 Paradegmata. 1) Ja prosdiorsoume touc x Z pou qoun thn idithta 9x 12 mod 15. 'Eqoume d = (9, 15) = 3 kai 3|12. 'Ara uprqoun akribc 3 klseic upolopwn mod 15 pou enai lseic. Gia na brome mia lsh thc ant- stoiqhc Diofantikc exswshc 9x15y = 12 efarmzoume ton Eukledeio algrijmo 15 = 9 1 + 6 9 = 1 6 + 3 6 = 2 3. 'Ara 3 = 961 = 9(1591) = 92151 kai pollaplasizontac me 4 qoume 9 8 15 4 = 12. Jtoume (x0, y0) = (8, 4). Ap thn apdeixh tou Jewrmatoc 1.5.3, isthta (4), lec oi lseic thc arqikc isotimac enai x 8 mod 15, x 3 mod 15, x 13 mod 15. 2) Mporome na lsoume thn isotima 7x 22 mod 10 me thn prohgome- nh mjodo enallaktik na parathrsoume ti afo (7, 10) = 1, to 7 enai antistryimo modulo 10 (Prtash 1.4.5), stw 7a 1 mod 10. Pollaplasizontac thn arqik isotima me a parnoume x 22a mod 10. Me ton Eukledeio algrijmo ( me poion llo trpo jloume) brskou- me to antstrofo tou 7 modulo 10, a 3 mod 10. 'Ara x 66 6 mod 10 (dete kai thn Efarmog met thn Prtash 1.4.5). Shmewsh Sto Pardeigma 1.5.4 1) mporome na ergastome kai wc exc. H dojesa isotima enai isodnamh me thn 3x 4 mod 5, pou qei akribc ma lsh ap to Jerhma 1.5.3, thn x 3 mod 5. Pc sqetzetai aut h lsh me tic treic lseic pou brkame sto Pardeigma 1.5.4 1); An sumbolsoume thn klsh upolopwn mod m tou a me [a]m, tte enai ekolo na dome ti [3]5 = [3]15 [8]15 [13]15. Me lla lgia blpoume ti qoume do perigrafc tou idou sunlou, dhlad tou {x Z|9x 12 mod 15}. Pio genik, mpore na apodeiqje to exc. 'Estw d, m jetiko akraioi ttoioi ste d|m. Tte gia kje akraio r qoume thn xnh nwsh [r]d = [r]m [r + d]m r + m d 1 d m . H apdeixh afnetai san skhsh.
  57. 57. 1.5. Diofantikc Exisseic kai Isotimec 57 Sustmata isotimin Sth sunqeia ja jewrsoume sustmata isotimin. 'Otan lme lsh enc sustmatoc isotimin ennoome kje akraio pou ikanopoie lec tic isotimec tou sustmatoc. Arqik parathrome ti enai dunatn na ssthma na mhn qei lsh an kai kje isotima tou sustmatoc qei lsh. Ac jewrsoume, gia pardeigma, to ssthma x 1 mod 2 x 2 mod 6. Aut den qei lsh, giat ap thn prth isotima parnoume 2|x 1 kai ap th deterh parnoume 2|x 2 kai epomnwc 2|1. Sthn perptwsh pou uprqoun akraioi x, x pou ikanopoion to ssthma x a mod m x b mod n parnoume x x mod m kai x x mod n, dhlad m|x x kai n|x x , kai epomnwc e|x x , pou e = (m, n). Sunepc, an uprqei lsh, aut enai monadik mod e. Sthn eidik perptwsh pou isqei (m, n) = 1, tte mporome na dexoume ti to prohgomeno ssthma qei lsh (kai enai monadik mod e, dhlad mod (mn)). Sqetik enai to paraktw Jerhma. 1.5.5 Jerhma (Kinezik Jerhma Upolopwn). 'Estw m1, m2, . . . , mr an do sqetik prtoi jetiko akraioi. Tte to ssthma isotimin x a1 mod m1 ... x ar mod mr qei monadik lsh modulo M = m1m2 . . . mr. Apdeixh. 'Uparxh: Jtoume Mk = M mk kai parathrome ti, lgw thc u- pjeshc, isqei (Mk, mk) = 1. Epomnwc to Mk enai antistryimo modulo mk 'Estw Mkyk 1 mod mk. Jtoume x = a1M1y1 + + arMryr (5) Epeid gia i = k qoume mi|Mk h (5) dnei x akMkyk mod mk, k = 1, 2, . . . , r. 'Ara x ak mod mk, k = 1, 2, . . . , r.
  58. 58. 58 1 Akraioi Monadikthta: 'Estw x kai x do lseic tou arqiko sustmatoc. Tte gia kje k = 1, 2, . . . , r qoume x x mod mk, dhlad mk|x x . Ap to Par- deigma 1.2.8 2) sumperanoume ti M|x x , afo oi mk enai an do sqetik prtoi. Tonzoume ed, ti to snolo twn akerawn pou enai lseic tou sustmatoc tou prohgoumnou Jewrmatoc enai h tom twn klsewn [a1]m1 , . . . , [ar]mr . 1.5.6 Paradegmata. 1) Gia na lsoume to ssthma x 1 mod 3 x 2 mod 5 x 3 mod 7 jtoume, smfwna me thn apdeixh tou Jewrmatoc 1.5.5, M = 3 5 7 = 105, M1 = 105 3 = 35, M2 = 105 5 = 21 kai M3 = 105 7 = 15. Gia na brome to y1 lnoume thn isotima 35y1 1 mod 3, dhlad thn 2y1 1 mod 3. 'Eqoume y1 2 mod 3. Brskoume to y2 ap thn 21y2 1 mod 5. 'Eqoume y2 1 mod 5. Brskoume to y3 ap thn 15y3 1 mod 7. 'Eqoume y3 1 mod 7. Telik x 1 35 2 + 2 21 1 + 3 15 1 mod 105, dhlad x 157 52 mod 105. 2) Mporome suqn na lsoume sustmata isotimin kai me diadoqikc a- ntikatastseic. (Gia th mjodo aut den qreizontai ta moduli mk na enai an do sqetik prtoi, all sthn perptwsh aut den gnwrzoume ek twn protrwn an uprqei lsh). Gia pardeigma stw x 1 mod 5 x 2 mod 6 x 3 mod 7 Gnwrzoume ap to Kinezik Jerhma Upolopwn ti to ssthma qei monadik lsh mod 210 thn opoa mporome na brome wc exc. Ap thn prth isotima qoume x = 5t + 1, t Z. Antikajistntac sth deterh qoume 5t 1 mod 6, thn opoa lnoume gia na brome t 5 mod 6. 'Ara t = 6u + 5, u Z. Sunepc x = 5(6u + 5) + 1 = 30u + 26.