Moswavlis biblioTeka j e m a L k I k n a z e
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(2) roca n = 1, 2, 3, 4, ... 40, igi iZleva martiv ricxvebs (ramdenime Sen TviTon Seamowme). ma- gram roca n = 41-s _ ara. Seamowme damoukide- blad. aseve formula: ϕϕϕϕϕ (n) = n 2 - 79n + 1601 (3) roca n = 1, 2, 3, 4, ...79, iZleva martiv ricx- vebs, xolo roca n = 80 _ ara. SegiZlia Seamow- mo.
dReisaTvis cnobili yvelaze didi martivi ricxvia:
2 86243 _ 1. am ricxvSi 25962 cifria! Tanamedrove kompiuteriTac ki didi mar- tivi ricxvebis gamoTvla Znelia da moiTxovs did dros.
_1 es martivi ricxvi rom vipovoT 2- iani Tavis Tavze 86 243-jer unda gavamravloT da Sedegs 1 gamovakloT. es Zalian didi ricx- via! marTlac warmoidgine, rom wertilma pirv- el dRes gaiara 2 mm da yovel momdevno dRes mis mier gavlili manZili orkecdeba. ra man- Zils gaivlis is: a) 5 dReSi? 2 5 =2 ×2×2×2×2=32 mm b) erT TveSi? (2 30 mm)
g) or TveSi? (2 60 mm)
d) erT weliwadSi? (2 365 mm)
2 mm 1 dReSi
2 dReSi 3 dReSi
4 dReSi 5 dReSi
(viciT mzidan dedamiwamde manZili daax- loebiT 150 000 000 km-ia). dReisaTvis cnobili yvelaze didi martivi ricxvis sidide rom vipovoT, es igivea vupasu- xoT kiTxvas: ra manZili eqneba gavlili ganx- ilul wertils 86 243 dReRamis Semdeg? sainteresoa, rom 86 243 dReRame ufro metia vidre 236 weliwadi da ganxiluli wertili gava CvenTvis cnobili samyaros sazRvrebs gareT gacilebiT adre, vidre saZie- bel manZils gaivlides! cota ram istoriidan: 1742 wels peter- 31 ricxvebis idumali samyaro burgis mecnierebaTa akademiis wevrma x. gold- baxma imave akademiis wevrs eilers miswera werili, sadac Camoayaliba erTi saintereso hipoTeza: nebismieri 5 -ze meti naturaluri ricxvi warmoadgens sami martivi ricx- vis jams . eilerma upasuxa, rom mas ar SeuZlia am hipoTezis damtkiceba, magram SeuZlia sxva hipoTezis Camoyalibeba, romelic dakavSirebu- lia pirvelTan: yoveli
2 -ze meti luwi naturaluri ricxvi warmoadgens ori martivi ricx- vis jams. 200 wlis ganmavlobaSi mravali qveynis cno- bili maTematikosebi cdilobdnen goldbax-eil- eris hipoTezebis damtkicebas, magram uSede- god. amave dros XX saukunis dasawyisSi am Tvisebis Semowmebam 9 000 000-mde aCvena, rom goldbaxis hipoTeza sworia. 1931 wels axal- gazrda rusma maTematikosma lev Snirelmanma mniSvnelovan warmatebas miaRwia am problemis gadawyvetaSi da bolos 1937 w. akademikosma i. m. vinogradovma TiTqmis bolomdis amoxsna es problema, magram sruli damtkiceba jer aravis mouxdenia! Semdegi problema, romelic goldbaxis prob- lemaze gacilebiT sainteresoa, Tavis gadawyve- tas jerjerobiT mcirediTac ver miuaxlovda. SemCneuli iyo, rom martivi ricxvebi araiS- viaTad gvxvdebian P da P+2 wyvilebis saxiT (aris erTi gamonaklisi 2 da 3): (3, 5); (5, 7); (11, 13); da a.S. aseT martiv ricxvebs tyupebi hqvia. ai, sakmaod didi tyupi ricxvebis wyvili: (1000000009649, 1000000009651). gamoTqmuli iyo da sarwmunod iTvleba hipoTe- za: tyupi ricxvebis simravle usasru- l o a . magram dRemde ar aris miRweuli araviTari warmateba am hipoTezis damtkicebaSi.
L
daakvirdi piTagoras cxrilSi gamoyofil kuTxeebs. daamtkice, rom am kuTxeebSi moTavsebuli ricxvebis jami qmnian kubebis mimdevrobebs: 1 3 , 2 3 , 3 3 , 4 3 , 5 3 , ... 2 L
ras udris 25-e kuTxeSi moTavsebuli ricxvebis jami? 1 cxadia 1+2+3+4+5+6+7+8 ×9=100. qvemoT moyvanil magaliTebSi tolobis marcxena mxares Camoweril ricxvebs Soris Casvi saWiro maTematikuri moqmedebis niSnebi. ise rom tolobebi WeSmariti gaxdes:
romeli ricxvi modis Semdeg? 3 L
aRmoaCine romeli wesis mixedviTaa mimdevrobis wevrebi SerCeuli: a) 11, 31, 41, 61, 71, 101, 131, ... b) 13, 23, 43, 53, ...
L Seamowme goldbaxisa da eileris hipoTezebi ramdenime ricxvisaTvis. 5 L aba, scade da gaarkvie ramdeni tyupi ricxvia pirvel aseulSi? 6 L ipove yvela martivi ricxvi, romelic erTdroulad aris ori martivi ricxvis jami da sxvaoba. 7 L ipove yvela martivi ricxvi, romelic miiReba gamosaxulebiT: 2 1 (n - 1)(n + 2), sadac n naturaluri ricxvia. 8 L daamtkice, rom namravli 1 .2.3.4.5. ... .19 (mokled iwereba 19! da ikiTxeba `19-is faqtoriali ~) iyofa 820 125-ze. Tavsatexi 32 ricxvebis idumali samyaro nikomax geraski (Zv. w. I s.) piTagoreli, cnobili berZeni filosofosi da maTematikosi werda: `srulyofili ricxvebi lamazia, magram lamazi nivTebi iSviaTia da araa bevri, uSno ki gvxvdeba mravlad. uamravia TiTqmis yvela ricxvi, maSin roca srulyofili ricxvebi cotaa ~. Zvelma berZnebma icodnen mxolod ori srulyofili ricxvi: 6 da 28. ra aqvT maT saerTo da riT gamoirCevian sxva ricxvebisagan? aba, ipove maTi gamyofebi da Semdeg Sekribe isini. miiReb igive ricxvebs: 6=1+2+3, 28=1+2+4+7+14. maSasadame, TiToeuli maTgani udris Tavisi sakuTari gamyofebis jams. aseTi Tvisebis mqone ricxvebs srulyofils uwodeben. evklidemde (Zv. w. III s.) mxolod es ori srulyofili ricxvi iyo cnobili. evklidem moifiqra formula:
sadac 2 P _ 1 martivi ricxvia. am formulis gamoyenebiT man kidev ori srulyofili ricxvi aRmoaCina:
mexuTe srulyifili ricxvia: 33 550 336, meeqvse _
meSvide _ 137 438 691 328; merve _ 2 305 843 008 139 952 128. mecxre srulyofili ricxvi gamoTvlili iyo mxolod 1883 wels rusi soflis mRvdlis i. v. perkuSinis mier. am ricxvSi aris 37 cifri.
×2 60 , sadac
(misi Sroma gmirobis tol- fasi iyo). cxadia aseTi ricxvebis xeliT angariSi Zalze Znelia (TiTqmis SeuZlebelia). meoce saukunis dasawyisSi gamoiyenes meqanikuri saTvleli manqanebi da 1911 wels aRmoaCines meaTe srulyofili ricxvi:
×2 88 , sadac
am ricxvSi aris 54 cifri. 1914 wels aRmoaCines meTerTmete (65 cifriani) srulyofili ricxvi:
×2 106 da meTormete (77 cifriani) 2 126 (2 127 _1) srulyofili ricxvi. SemdegSi gamoigones eleqtronuli saTvleli manqanebi da 1952 wlis 30 ianvars amerikelma maTematikosma robinsonma, kaliforniis univer- sitetSi gamoiyena eleqtronuli saangariSo manqana 2
ricxvis simartivis Sesamowmeblad. man Tavdapirvelad gadawyvita 2
ricxvis
simartivis Semowmeba da miiwvia lemeri, romelmac 20 wlis win TiTqmis erTi weli moandoma amis angariSs. lemerma miiRo didi siamovneba, rodesac dainaxa, rom manqanam miiRo igive Sedegi 18 wamSi. imisaTvis, rom mieRoT Semdegi srulyofili ricxvi, manqanam daiwyo axali martivi ricxvebis Zebna. man or saaTSi 42 ricxvi Seamowma, romelTa Soris yvelaze mcires hqonda 80-is toli cifrTa ricxvi! yvela es ricxvi aRmoCnda Sedgenili. axali (mecamete) srulyofili ricxvi manqanam aRmoaCina 1952 wlis 30 ianvars saRamos:
am ricxvSi 314 cifria. imave dRis SuaRamisas manqanam aRmoaCina meToTxmete srulyofili ricxvi. Seamowma kidev 13 evklidiseuli ricxvi, da man miagno martiv ricxvs 2 607 _1 , romelsac aqvs 183 cifri (aTobiT sistemaSi). es srulyofili ricxvia:
meToTxmete srulyofil ricxvs aqvs 366 cifri. 1957 wlis seqtemberSi Svedi maTematikosis g. rizelis mier napovni iqna meTvrmete srulyofili ricxvi. eleqtronul-saangariSo manqanis gamoyenebiT man xuT naxevar saaTSi daadgina 2 3217 _1 ricxvis martivoba da miiRo meTvramete srulyofili ricxvi:
masSi daaxloebiT 2000 cifria. Semdegi srulyofili ricxvebis Zebna moiTxovs ufro meti da meti moculobis angariSs. saangariSo teqnika TandaTanobiT srulyofili xdeba da 1962 w. moZebnili iqna ori axali srulyofili ricxvi, 1965 wels kidev sami. am ricxvebs evklidis formulaSi Seesabameba P = 4253, 4423, 9689, 9941 da 11213. srulyofil ricxvSi 2 11212 (2 11213 _1) aris 3376 s s
s srulyofili r rulyofili r rulyofili r rulyofili r rulyofili ricxvebi. icxvebi. icxvebi.
icxvebi. icxvebi.
angar angar
angar angar
angariSi da saangar iSi da saangar iSi da saangar iSi da saangar iSi da saangariSoebi iSoebi
iSoebi iSoebi
iSoebi 33 ricxvebis idumali samyaro cifri! udidesi maTematikosi edmund landau aR- n i S n a v s : `... dRemdis ori problemaa gadauWreli: usasruloa Tu ara luwi srulyofil ricxvTa simravle? _ ar v i c i . _ a r s e b o b s T u a r a k e n t i s r u l y o f i l r i c x v T a u s a s r u l o s i m r a v l e ? _ m e i s i c k i a r v i c i arsebobs Tu ara erTi mainc aseTi~. cxadia iseTi damxmaris saSualebiT, rogoricaa saangariSo manqanebi adamianma SeZlo aseTi veberTela srulyofili ricxvebis dadgena. dReisaTvis cnobilia 30-ze meti srulyofili ricxvi romelTa Soris udidess Seesabameba:
srulyofili ricxvebis Zebnis istoria, TvalnaTliv gviCvenebs, Tu rogor zrdis saan- gariSo manqanebi adamianis SesaZleblobebs. maTematikosebi yovelTvis ocnebobdnen iseT daxmarebaze, romelic gaamartivebda maT Sromas xangrZlivi da damRleli gamoTvlebisagan. erT-erTi gamomTvleli mowyobiloba iyo abaki. es saangariSo CarCo dResac gamoiyeneba zogierT qveyanaSi. logariTmuli cxrilebi XVII saukunis dasawyisSi daamuSava jon neperma da aadvilebda gamoTvlebs TiTqmis 4 saukunis ganmavlobaSi. msoflioSi pirveli saTvlel-saangariSo mowyobiloba (manqana) aago 19 wlis blez paskalma 1642 wels (sur. 1). dRes TiToeul Tqvengans aqvs saSualeba gamoiyenos miniaturuli saangariSo manqana- mikrokalkulatori (sur. 2), romelic saSualebas iZleva gamoTvlebi Catardes swrafad. igi namdvilad iqca maTematikosebis (da ara marto maTTvis) megobrad da mxsnelad. mikrokalkulatorze, Tu daakvirdebi ric- xvebis klaviSebis gan- lagebas Zalian advilad aRmoaCen saintereso ricxviT kanonzomiere- bebs: romeliRaca ric- xvebis sxvaoba yovel- Tvis 111-is an 333-is tolia. aba, moZebne, romelia es ricxvebi? a g r e T v e r o m e - liRaca orniSna ric- xvebis sxvaoba yovel- Tvis 27, 198 an 594-is tolia. zogierTi jami yovelTvis iyofa 11-ze. maTi moZebna ar gagi- Wirdeba, Tu gulmod- gined daakvirdebi da sxvadasxva variants Sea- mowmeb. klaviSebze kidev sxva saintereso kanon- zomierebebis aRmoCenac SeiZleba! ivarjiSe da aRmoaCine! magaliTad: 3+7=10 1+5+9=15
1+9=10 3+5+7=15
2+8=10 2+5+8=15
4+6=10 4+5+6=15
1 L dawere kent ricxvTa sasruli mimdevroba 1-dan 999-mde. a) ras udris am ricxvebis jami? b) Seadgine analogiuri amocana luwi ricxvebisa da naturaluri ricxvebis SemTxvevaSi.
L moZebne da gamoiyene yvelaze martivi angariSis wesi Semdegi magaliTebisaTvis: a) 9 9 _ 9 7 + 9 5 _ 9 3 + 9 1 _ 8 9 + . . . + 7 _ 5 + 3 _ 1 . b)
1 1 1 1 1 1 1 1 1 . 1 2 2 3
3 4 4 5
5 6 6 7
7 8 8 9
9 10 + + + + + + + + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 3
L a) swrafad ipove 1 . 2.3.4.5 .6+1 ricxvis 5-ze gayofisas miRebuli ganayofi da naSTi.
b) ipove zepirad 1000000–(1000000–{1000000–[1000000-(1000000-999999]}). 4 L frangma maTematikosma sofi Jermenma (1776-1831), roca moswavle iyo, daamtkica, rom K 4 + 4, sadac K erTze meti nebismieri mTeli ricxvia, aris Sedgenili. Sen daamtkiceb? sur. 1 sur. 2
3. j. kiknaZe 198:9=22. 594:9=66.
34 ricxvebis idumali samyaro cnobili Sveicarieli maTematikosebis, bernulebis ojaxis warmomadgenelma iohan III bernulma 1773 wels berlinis akademiis SromebSi gamoaqveyna cxrili, sadac mocemuli iyo n raodenobis erTianebiT dawerili ricxvebis martivi gamyofebi: miuxedavad imisa, rom man ver SeZlo am saxis zogierTi ricxvebis martivi gamyofebis moZebna (n=11, 17, 29), xolo sami ricxvis (n=20, 25, 27) gamyofebi ar iyo dayvanili martiv gamyofebamde, miuxedavad mis mier daSvebuli Secdomebisa (n=22,
gigantur Sromas am didi ricxvebis martivi gamyofebis moZebnisaTvis. cxrilSi bernulim varskvlaviT aRniSna misTvis saeWvo SemTxvevebi. am cxrilis gamoqveynebidan TiTqmis 200 wlis Semdeg, 1964 wels niu-iorkSi a. beilerma gamoaqveyna wigni `saxaliso ricxvTa Teoria~ sadac am ricxvebs miuZRvna mTeli Tavi saxelwodebiT `111 ... 1111~, da Semoitana maTTvis termini `repiunitebi~ (`repunit~, Semokleba inglisuri sityvebisa repeated unit _ gameorebuli erTiani). es termini qarTul cnobarebSi jerjerobiT aRniSnuli ar aris. aTobiT sistemaSi repiunitebs aRniSnaven R-iT. mag. R 1 = 1, R 2 = 11 da a.S. maTematikosebi muSaoben am cxrilis gafar- Toebaze da 1975 wlisaTvis cxrilSi n-ma miaRwia 3000 (s. eits), magram masSi sakmao raodenobis uzustobebia (amJamad am uzustobebis nawili likvidirebulia da yvela n = 162 ricxvisaTvis moZebnilia martivi gamyofebi). `111 ... 1111~ _ repiunitebi gansakuTrebul interess iwvevs martivi repiunitebi. ukve damtkicebulia, rom R 1 , R 19 (1918 w.), R 23 (1929 w.), R 317 (1978 w.) da R 1037 (1985 w.) martivi ricxvebia. e.i. repiunitebis ojaxidan jer-jerobiT cnobilia mxolod xuTi martivi ricxvi. aRmoCenilia repiunitebis mravali saintereso Tviseba, albaT bevri jer kidev ucnobia. namravli R
×××××R j (-i, -Ji indeqsebia _ natu- raluri ricxvebia), rodesac
≥ ≥ 9 , war-
moadgens palindromul ricxvs (12... j ... 21 saxisas). ricxvi palindromi ewodeba naturalur ricxvs, romlis Canaweri emTxveva misi sarkuli anareklis Canawers. magaliTebi: 11111 ×××××111 = 1233321, 111 2 = 12321. Tu
9 > ≥ j i , maSin R i ×××××R j namravli ar aris palindromi.
35 ricxvebis idumali samyaro 1 L romeli ricxvebiT SeiZleba Sec- valo asoebi, rom cxra Sesak- rebTa jami gaxdes repiuniti? 3 L romeli ori repiunitis namravlia ricxvi 123455554321? 4 L avtomobilebis gayidvis Semdeg firmis Semosavali gaxda 1111111 dolari. ramdeni avtomo- bili gauyidia firmas, Tu TiToeulis fasi erTnairi iyo? 5 L cxadia: 121 = 11 ×××××11, 11211 = 111 ×××××101, 1112111 = 1111 ×××××1001. e.i. palindromebi 121, 11211, 1112111 ar arian martivi ricxvebi. daamtkice, rom erTeuli
erTeuli n n 11 ... 1 2 1 ... 11 repiuniti nebismieri n-isaTvis ar aris martivi ricxvi. 6 L SegiZlia oTxi 2-ianiT da Svidi 5-ianiT miiRo repiunitebi? 7 L Tu 12 345 679 gaamravleb 9-ze, miiReb repiunits (Seamowme). romel ricxvze unda gaamrav- lo mocemuli ricxvi, rom miiRo a) mxolod xuTianebiT da b) mxolod cxrianebiT dawe- rili ricxvebi. 8 L dawere formula, romliTac miiReb repiunitebs. ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` + 2 L romeli ricxvebiT Secvlidi asoebs? R R R R R R R R R R R R R R ×××××
REPUNITINUPER Ð Å Ï Ü Þ Í È Ò 36 ricxvebis idumali samyaro `zepiri angariSi~ _ cnobili rusi mxatvris bogdanov-belinskis farTod cnobili namuSevrebidan erT-erTia, romelic moskovis tretiakovis galereaSi inaxeba. mxatvarma daxata Tavisi skolis erTi gakveTili da maswavleblis rolSi warmoadgina cnobili pedagogi s. a. raCinski. kargad daakvirdi suraTs. masze Sen aRmoaCen Zveli saxalxo skolis moswavleebis cxovrebas maTematikis gakveTilze. dainaxav maT sxvadasxva saxes, xasiaTs, maZiebel gonebas, sazrianobas da niWierebas. mxatvari gadmoscems maT gatacebas skoliT da amasTan erTad maswavleblis unars daainteresos bavSvebi ... magaliTi, romlis amoxsniTac isini gatacebuli arian dafaze weria. ai isic: Sen mas zepirad amoxsni? Yüklə 4,69 Mb. Dostları ilə paylaş:
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