fibonachi kram

ganitat, fibonachi kram haah eka kram aahe jyamadhye pratyek sankhya hii aadhichya done sankhyanchi berries asate. fibonachi anukramancha bhag asalelya sankhyanna fibonachi sankhya mhanatat, jee samanyatah darshavili jaatat. jaree kahee lekhak 1 aanee 1 pasoon anukram suru karatat kinwa kadhikadhi (phybonachipramanech) 1 aanee 2 pasoon. 0 aanee 1 pasoon suru hott asale, taree kramatil pahili kahee moolye ashee aaheth
done lambichya shabdanshanpasun tayar jhalelya sanskrut kavitechya sambhavya namunyanchi ganana karnyavar pingalane kelelya karyat, ii. sha. puu. 200 chya suruvatilach, bharatiy ganitat fibonachi sankhyanche varnan pahilyanda kele gele. tyanna italian ganitshastradnya leonardo auph pisa, jyala fibonachi asehi mhanatat, tyachya 1202 chya 'lieber abachi' yaa pustakat paschim european ganitacha kram sadar kelyapasoon tyanche naav denyat aale aahe.
ganitat fibonachi sankhya anpekshitpane anekada itaki disun yete qee tyanchya abhyasasathi samarpit eka sampoorn niyatkalik aahe, 'fibonachi traimasik'. fibonachi sankhyanchya anuprayoganmadhye fibonachi shodh tantra aanee fibonachi heep deta structure yasarkhe sanganak algoridam aanee samantar aanee vitarit pranalinna ekmekanshi jodanyasathi vaparale jaanare fibonachi cubes navache aalekh samavisht aahet. zadanmadhye fandya padane, khodavar pananchi mandani, ananasache phal ankurane, kelyachi phule yene aanee devdar shankuchya kondyanchi mandani yasarkhya jaivik paristhitinmadhye dekhil teey disatat, jaree teey serve prajatinmadhye aadhalat nasale taree.
fibonachi sankhya dekhil suvarn gunottarasah drudhpane sambandhit aaheh binnetche sootra th fibonachi sankhya aanee suvarn gunottar yaa drushtine vyakt karate aanee asey soochit karate qee salag done fibonachi sankhyanche gunottar vadhatana suvarn gunottarakade valate. fibonachi sankhya dekhil lucas sankhyanshi jawalun sambandhit aahet, jyaa samaan punaravrutti sambandhanche palan karatat aanee fibonachi sankhyansah lucas anukramanchi purak jodee tayar karatat.
vyakhya.
fibonachi sankhya punaravrutti sambandhadvare paribhashit kelee jaau shakate
sootra _ 1
aanee
sootra _ 2
sathee.
kahee junya vyakhyannusar, moolya sootra _ 3 vagalale jaate, jenekarun kram sootra _ 4 neey suru hoto aanee punaravrutti sootra _ 5 sathee vaidh aahe. 
pahilya 20 fibonachi sankhya pudhilpramane aaheth
itihas.
bharat.
bharatiy ganitat, sanskrut gadyachya sandarbhat, fibonachi kram aadhalato. sanskrut kavyatmak paramparet, 2 ekkanchya kalavadhichya lamb (ell) aksharanchya serve namunyanchi ganana karnyat rus hota, jyaa 1 ekkachya kalavadhichya lahan (ess) aksharansah jodalelya hotya. dilelya ekoon kalavadhisah salag l aanee s chya vegvegalya namunyanchi ganana kelyane fibonachi sankhya milateh kalavadhichya ekkanchya namunyanchi sankhya aahe.
fibonachi kramache dnyan pingala (ii. sha. puu. 450-ii. sha. puu. 200) pasoon vyakt kele gele. sinh yanni pingalache goodh sootra 'misrao chaa' ('donhi mishra aahet') uddhrut kele aahe aanee jyaa vidvananni tyacha sandarbh assa deela aahe qee, talansathichya namunyanchi sankhya prakarananmadhye eka [s] aanee prakarananmadhye eka [l] jodoon prapt hote. bharat muni dekhil natyashastrat (see. ii. sha. puu. 100. 350 edee).
tathapi, yaa kramache sarvat spasht spashtikaran virhankchya karyat udbhavate (ii. ii. sha. 700), jyanche swatache kaam haravale aahe, parantu gopal (ii. 1135):
aadhichya done meterche farak [haah farak aahe]... udaharanarth, chaar meater lambisathi, done meater aanee teen meater misalale gelyane paach farak hotat. [8,13,21 hii udaharane dilee aahet]... asha prakare, serve 'matru-vrutta' [gadya sanyojan] madhye yaa prakriyeche palan kele pahije.
hemchandra (ii. 1150) laa anukramachya dnyanache shrey dekhil dile jaate, asey lihitana qee "shevatchya aanee shevatchya aadhi asalelya ekachi berries hii sankhya aahe... pudhil matru-vruttache."
europe.
fibonachi kram pratham fibonachichya 'lieber abasi' (gananeche pustak, 1202) yaa pustakat aadhalato, jithe sasachya loksankhyechya vadhichi ganana karnyasathi tyacha vapar keela jaato. fibonachi sasachya aadarsh (jaivikdrushtya avastav) loksankhyechya vadhicha vichar karate, asey gruheet dharunah sasachya navyane janmalelya prajnan jodila shetat thewale jaate; pratyek prajnan jodee eka mahinyachya vayat sangopan karate aanee tyanchya gunankatoon mahinyachya shevati teey nehmich sasachya aanakhi eka jodiche utpadan karatat; aanee sase kadhihi marat naheet, parantu kayamche prajnan karat rahatat. fibonachine eka kode mandaleh eka varshat kitee jodya asateel?
mahinyachya shevati sasachya jodyanchi sankhya hii paripakv jodyanchi sankhya (mhanje mahinyatil jodyanchi sankhya) aanee gelya mahinyat (mahina) jiwant asalelya jodyanchi sankhya itaki asate. dahavya mahinyatil sankhya hii dahavya fibonachi sankhya aahe.
fibonachi anukram' heey naav pahilyanda pahilyanda shatkatil sankhya siddhantkar edward lucas yanni vaparale hote.
suvarn gunottaranshi sambandh.
bund swarupachi abhivyakti.
sthir gunank asalelya resheey punaravruttidvare paribhashit kelelya pratyek kramanusar, fibonachi sankhyanchi abhivyakti bund swarupachi asate. jaree teey aadhich abraham dii moivar aanee daniele bernauli yanna maheet hote, taree french ganitshastradnya jacks phillip merrie binnet yanchya navavarun tyala binnetche sootra mhanoon olakhale jaau lagale aahe.
sootra _ 6
kuthe?
sootra _ 7
heey suvarn gunottar aahe aanee tyache sanyugma aaheh
sootra _ 8
sootra _ 9 pasoon, heey sootra asey dekhil lihile jaau shakate
sootra _ 10
kram aanee yaa sthirankanmadhil sambandh pahanyasathi, lakshat ghya qee aanee heey samikaran sootra _ 11 chey donhi upay aahet aanee asha prakare sootra _ 12 mhanje fibonachi punaravruttichya aanee samadhanachya shakti. gunankatoon shabdant,
sootra _ 13
heey konatyahi mulyansathi aanee, dware paribhashit kelelya kramache anusaran karate
sootra _ 14
teech punaravrutti samadhankarak,
sootra _ 15
jarr aanee nivadle gele jenekarun aanee nantar parinami anukram fibonachi kram asava. heey samikaranachya pranalichi aavashyakta aanee samadhan karnyaitakech aaheh
sootra _ 16
jyacha upay aahe
sootra _ 17
aavashyak sootra tayar karane.
prarambhik moolye ghetalyas aanee aniyantrit sthirank honyasathi, eka adhik samanya upay aaheh
sootra _ 18
kuthe?
sootra _ 19
goal karoon ganana.
tevhapasoon
sootra _ 20 heey sarvansathi, sankhya hii sootra _ 21 chee sarvat jawalchi purnank aahe. mhanoon, teey sarvat jawalche purnank karya vaparoon, goal karoon shodhale jaau shakateh
sootra _ 22
kharey tarr, goal truti khoop lahan aahe, 0.01 sathee 0.00 peksha qamee aanee 0.01 sathee 0.01 peksha qamee aahe. fibonachi kramankacha nirdeshank shodhanyasathi heey sootra sahajpane ulate kele jaateh
sootra _ 23
tyaeivaji flore function vaparnyane fibonachi kramankacha sarvat motha nirdeshank midato joou yapeksha motha nasatoh
sootra _ 24
jethe sootra _ 25, sootra _ 26 aanee sootra _ 27.
pariman.
fn" heey sootra _ 28 sathee asymptotic asalyane, "f" n "madheel ankanchi sankhya sootra _ 29 sathee asymptotic aahe. parinami, pratyek purnank" d "> 1 sathee" d "dashansh ankansah ektar 4 kinwa 5 fibonachi sankhya aahet.
adhik samanyatah, aadhar "buub" pratinidhitvamadhye, "eff" aine "madheel ankanchi sankhya sootra _ 30 sathee asymptotic aahe.
salag bhaganchi maryada.
johann keplarne nirikshan kele qee salag fibonachi sankhyanche gunottar ektrit hote. tyanni lihile qee "5 mhanje 8 mhanoon 8 teey 13, vyavharikdrushtya, aanee 8 mhanje 13 mhanoon, 13 teey 21 jawaljawal aahe", aanee assa nishkarsh kadhala qee heey gunottar suvarn gunottar sutrakade jaatat. 
sootra _ 32
sootra _ 33 aanee sootra _ 34 yaa prarambhik mulyanchi parva naan karata, heey abhisaran sootra _ 35 paryant tikate. heey binnetche sootra vaparoon satyapit kele jaau shakate. udaharanarth, 3 aanee 2 hii prarambhik moolye 3,2,5,7,12,19,31,50,81,131,212,343,555, haah kram tayar karatat... . yaa kramatil salag sandnyanche gunottar suvarn gunottarakade samaan abhisaran darshavate.
sarvasadharanpane, sootra _ 36, kaaran salag fibonachi sankhyanmadhil gunottar sootra _ 37 paryant pohochate.
adhikaranche vighatan.
kaaran suvarn gunottar samikaranache samadhan karate
sootra _ 38
yaa abhivyakticha vapar uchch shaktinche sootra _ 39 heey qamee shaktinche resheey karya mhanoon vighatit karnyasathi keela jaau shakato, jeey tyaulat sootra _ 37 aanee 1 chya resheey sanyogaparyant vighatit kele jaau shakate. parinami punaravrutti sambandhanmule resheey gunank mhanoon fibonachi sankhya milateh
sootra _ 41
heey samikaran khaleel goshtinvar preranadvare siddha kele jaau shakateh
sootra _ 42
sootra _ 43 sathee, heey sootra _ 44 dekhil aahe aanee asehi aahe qee
sootra _ 45
jarr fibonachi kram "eff. aine." haah fibonachi niyam sootra _ 46 vaparoon nakaratmak purnankanparyant vistarit keela gela tarr hii abhivyakti dekhil kharee aahe.
olakh.
jarr aanee jarr sootra _ 47 kinwa sootra _ 48 madheel kiman eka paripoorn varg asel tarach, 'x' haah sakaratmak purnank fibonachi sankhya aahe yacha purava binnetche sootra dete. yache kaaran asey qee sootra _ 49 mhanoon lihita yenare binnetche sootra, sootra _ 50 neey gunale jaau shakate aanee sootra _ 39 madhye chaturbhuj sutradvare chaturbhuj samikaran mhanoon sodavale jaau shakateh
sootra _ 52
yachee tulana sootra _ 53 shhee kelyas, teey asey aahe qee
visheshtah, davya hatachi bajoo eka paripoorn chauras aahe.
matricks form.
resheey farak samikarananchi eka dvimiteey pranali jee fibonachi anukramache varnan karate
sootra _ 55
paryayipane soochit
sootra _ 56
jeey sootra _ 57 dete. matrixchi eigenvalues sootra _ 58 aanee sootra _ 59 sambandhit igenvectors sambandhit aahet.
sootra _ 60
aanee
sootra _ 61
prarambhik moolya asey aahe
sootra _ 62
tyanantar hii sandnya aahe qee
sootra _ 63
yavarun, fibonachi maliketil tisra ghatak
bund swarupachi abhivyakti mhanoon thet vachata yeteh
sootra _ 64
samaan, teech ganana tyachya igendecomposition vaparadvare vikarnakaranadvare kelee jaau shakateh
sootra _ 65
jethe sootra _ 66 aanee sootra _ 67
tyamule fibonachi maliketil th ghatkasathi bund swarupachi abhivyakti dilee jaate
sootra _ 68
jeey punha utpann dete
sootra _ 69
matrixmadhye −1 chaa nirdharak asato aanee tyamule too 2 × 2 ekmudra matricks aahe.
suvarn gunottarasathichya akhand apurnank pratinidhitvanusar haah gunadharm samajala jaau shakatoh
sootra _ 70
fibonachi sankhya hii satatchya apurnankachya kramik abhisariche gunottar mhanoon udbhavate aanee konatyahi nirantar apurnankachya kramik abhisaripasun tayar jhalelya matrixmadhye + 1 kinwa-1 chaa nirdharak asato. matrixche pratinidhitva fibonachi sankhyansathi khaleel bund-swarupachi abhivyakti deteh
sootra _ 71
dilelya prakarasathi, heey matricks, ghatankadvare varg paddhaticha vapar karoon, ankganitiy karyanmadhye mojale jaau shakate.
yaa samikaranachya donhi bajuncha nirdharak ghetalyane casinichi olakh milate,
sootra _ 72
yavyatirikt, konatyahi chauras matrixsathi, khaleel olakh prapt kelee jaau shakate (teey matricks utpadanachya done vegvegalya gunankanmadhun prapt kele jaatat aanee eka sahajpane pahilya gunankatoon gunankatoon gunankamadhye badal karoon kadhu shakato),
sootra _ 73
visheshtah,
sootra _ 74
hii shevatchi done olakh, ankganitiy kriyanmadhye aanee velenusar, done ankanchya gunakarasathi vel kuthe aahe, fibonachi sankhya punaravrutti mhanoon mojanyacha marg pradan karatat. heey bund-form matricks sutratoon th fibonachi sankhya mojanyachya veleshi julate, parantu jarr konee aadhich ganana kelelya fibonachi sankhya (memoisations punaravrutti) punha mojane talhale tarr qamee anavashyak anavashyak.
sanyukt olakh.
sanyukt purave.
fibonachi sankhyancha samavesh asaleli bahutek olakh hii ektrit yuktivad vaparoon siddha kelee jaau shakate, kaaran sootra 75 chaa arth 1s aanee 2s chya (bahudha rikamya) anukramanchi sankhya mhanoon lavala jaau shakato, jyanchi berries sootra 76 aahe. hii sootra 75 chee vyakhya mhanoon parampara sootra 3 saha ghetali jaau shakate, mhanje assa konatahi kram astitvat nahee jyachi berries-1 aahe aanee sootra 79, mhanje rikama kram 0 paryant jodato. khalilpramane, sootra 80 hii sanchachi mukhyata aaheh
asha prakare punaravrutti sambandh
sootra _ 87
sootra _ 75 anukramanna done naun-overlapping sanchanmadhye vibhajit karoon samajale jaau shakate jethe serve anukram ektar 1 kinwa 2 neey suru hotatah
sootra _ 89
pahila ghatak vagaloon, pratyek kramatil urvarit sandnya sootra _ 90 kinwa sootra _ 91 chya berries karatat aanee pratyek sanchatil mukhyata sootra _ 92 kinwa sootra _ 93 aahe, jeey ekoon sootra _ 94 anukram dete, jeey darshavate qee heey sootra _ 75 chya barobariche aahe.
ashach prakare heey darshavile jaau shakate qee 'aine' vyaparyantchya pahilya fibonachi sankhyanchi berries ('aine' + 2) aanee fibonachi sankhya vaja 1 chya samaan aahe. chinhanmadhyeh
sootra _ 96
heey pahilya pahilya sthanavar aadharit sootra _ 97 madhye berries kelelya serve anukramanna vibhajit karoon pahile jaau shakate. visheshtah, pratyek sanchamadhye tyaa anukramancha samavesh asato jeey sootra _ 98 pasoon suru hotat aanee shevatchya done sanch sootra _ 99 paryant pratyeki cardinality 1 saha.
purvipramanech tyach tarkshastrache anusaran karoon, pratyek sanchachya mukhyatechi berries karoon aapan pahato qee
... jethe shevatchya done sandnyanche moolya sootra 79 aahe. yavarun teey sootra 102 chey anusaran karate.
samaan yuktivad, pahilya 2 aivaji pahilya 1 chya sthanane berries gatbaddh kelyane aanakhi done olakh milateh
sootra _ 103
aanee
sootra _ 104
shabdat sangayache tarr, sootra 105 paryant visham nirdeshank asalelya pahilya fibonachi sankhyanchi berries hii (2 'aine') vee fibonachi sankhya aahe aanee sootra 106 paryant summ nirdeshank asalelya pahilya fibonachi sankhyanchi berries hii (2 'aine' + 1) seent fibonachi sankhya vaja 1 aahe.
siddha karnyasathi eka vegali yukti vaparali jaau shakate
sootra _ 107
kinwa shabdat sangayache tarr, sootra _ 75 paryantchya pahilya fibonachi sankhyanchya varganchi berries hii 'aine' vya aanee ('aine' + 1) seent fibonachi sankhyanchi gunakar aahe. heey pahanyasathi, _ 109 yaa aakarachya sootrachya fibonachi aayatakrutipasun suruvat qaraa aanee tyala _ 110 yaa aakarachya sootrachya varganmadhye vighatit qaraa; yavarun kshetranchi tulana karoon olakh khalilpramane aaheh
pratikatmak paddhat.
anukramanche sootra _ 111 dekhil pratikatmak paddhat vaparoon manale jaate. adhik tantotantpane, haah kram vishisht sanyojan vargashi sambandhit aahe. yaa anukramache spashtikaran sootra 112 aahe. kharokhar, varr sangitalyapramane, sootra 113-vaa fibonachi sankhya 1 aanee 2 yaa sandnya vaparoon sootra 76 chya sanyukt rachananchya (krambaddha vibhajan) sankhyeitaki aahe.
heey asey aahe qee fibonachi anukramache samanya utpann karya, sootra _ 115, heey tarksangat karya sootra _ 116 aahe.
preran purave.
ganiteey preran vaparoon fibonachi olakh anekada sahajpane siddha kelee jaau shakate.
udaharanarth, punarvichar qaraa
sootra _ 117
donhi bajunna _ 118 heey sootra jodalyane milate
aanee mhanoon aamchyakade sootra _ 97 chey sootra aahe.
sootra _ 121
tyachpramane, donhi bajunna sootra _ 122 joda
sootra _ 107
dene
sootra _ 124
sootra _ 125
dwishir sootra purave.
dwishir sootra aahe
sootra _ 126
yacha vapar fibonachi olakh siddha karnyasathi keela jaau shakato.
udaharanarth, heey sootra siddha karnyasathi _ 127
lakshat ghya qee davya hatachi bajoo sutrane gunakar kelyas _ 128 banate
sootra _ 129
aavashyakta bhasalyas, samikarane sopee karnyasathi tathya sootra _ 130 aanee sootra _ 131 vapara.
itar olakh.
vividh paddhatincha vapar karoon itar anek olakh prapt kelee jaau shakatat. tyapaiki kahee yethe aaheth
cassini aanee catalanchi olakh.
casinichi olakh sangate qee
sootra _ 132
catalanchi olakh hii eka samanyikaran aaheh
sootra _ 133
dii 'okagnechi olakh.
sootra _ 134
sootra _ 135
jethe "l" "n" hii "n"-th lucas sankhya aahe. shevatchi olakh mhanje 'aine' duppat karnyasathichi olakh; yaa prakarchya itar olakh
sootra _ 136
casinichya olakhinusar.
sootra _ 137
sootra _ 138
sootra _ 139
heey jaali qamee karnyacha vapar karoon prayogikdrushtya aadhalu shakatat aanee fibonachi sankhya ghatak karnyasathi vishesh sankhya kshetra chalani sthapit karnyasathi upayukt aahet.
adhik sarvasadharanpane,
sootra _ 140
kinwa paryayi
sootra _ 141
heey sootra thevun, ekhadyala punha vareel vibhag matricks formchya shevatchi sutre midatat.
karya nirman karane.
fibonachi anukramache utpadan karya mhanje shakti malika
sootra _ 142
hii malika konatyahi jatil sankhya sutrasathi abhisari aahe _ 143 samadhankarak sootra _ 144 aanee tyachya bereejche eka sadhe bund swaroop aaheh
sootra _ 145
heey sootra _ 145 neey gunakar karoon siddha kele jaau shakateh
sootra _ 147
jethe formula _ 148 chaa samavesh asalelya formula _ 149 chya serve sandnya paribhashit fibonachi punaravrutti sambandhamule radd hotat.
aanshik ansh vighatan asey dile jaate
sootra _ 150
jethe sootra _ 151 heey suvarn gunottar aahe aanee sootra _ 152 heey tyache sanyugma aahe.
sambandhit karya sootra _ 153 heey negafibonachi sankhyansathi nirman karnare karya aahe aanee sootra _ 154 karyatmak samikaranache samadhan karate.
sootra _ 155
paraspar rakma.
paraspar fibonachi sankhyanvaril anant berries kadhikadhi theeta functionschya drushtine moolyankan kelee jaau shakate. udaharanarth, pratyek visham-anukramit paraspar fibonachi kramankachi berries ashee lihili jaau shakate
sootra _ 156
aanee vargabaddh paraspar fibonachi sankhyanchi berries
sootra _ 157
aanee vargakar fibonachi sankhyanchi eka 'nested' berries aahe jee suvarn gunottar paraspar dete,
sootra _ 158
serve summ-anukramit paraspar fibonachi sankhyanchi berries aahe
sootra _ 159
sootra 161 pasoon lambert malikechya sutrasah _ 160
tyamule paraspar fibonachi sthirank aahe
sootra _ 162
shivay, hii sankhya richard andrey-jenin yanni atarkik siddha kelee aahe.
millinchi malika olakh dete
sootra _ 163
jeey bund swarupat tyachya aanshik berijsathi khalilpramane aahe kaaran "aine" anantakade valateh
sootra _ 164
avibhajya aanee vibhajyata.
vibhajaniyata gunadharm.
anukramachi pratyek tisari sankhya summ (sootra _ 165 chee gunank) asate aanee adhik samanyatah, anukramachi pratyek "keey" vee sankhya "efke" chee gunank asate. asha prakare fibonachi kram heey vibhajyata kramache eka udaharan aahe. kharey tarr, fibonachi kram adhik majboot vibhajaniyata gunavisheshache samadhan karato
sootra _ 166
sarvat mothe samanya vibhajak karya kuthe aahe?
visheshtah, konatyahi salag teen fibonachi sankhya jodinusar saharth aahet kaaran sootra _ 167 aanee sootra _ 168. mhanje,
pratyek 'aine' sathee.
pratyek abhajya sankhya "pii" fibonachi sankhya vibhajit karate jee "pii" modulo 5 chya moolyane nirdharit kelee jaau shakate. jarr "pii" heey 1 kinwa 4 modulo 5 laa sarvasam asel tarr "pii" heey "eff" pii "-1 laa vibhajit karate aanee jarr" pii "heey 2 kinwa 3 modulo 5 laa sarvasam asel tarr," pii "heey" eff "pii" + 1 laa vibhajit karate. urvarit prakaran mhanje "pii" = 5, aanee yaa prakaranat "pii" heey "fp" laa vibhajit karate.
sootra _ 170
yaa prakarananna legender chinh vaparoon ekach, naun-peacewise sutrat ekatra kele jaau shakateh
sootra _ 171
prathamik chachani.
vareel sootra prathamik chachani mhanoon yaa arthane vaparale jaau shakate qee jarr
sootra _ 172
jethe legendre chinh jakobi chinhane badalale aahe, tevha haah purava aahe qee "aine" haah avibhajya aahe aanee jarr too dharoon thevanyat ayashasvi jhaala tarr "aine" nishchitpane avibhajya nahee. jarr 'aine' sammishra asel aanee sootra purn karat asel, tarr 'aine' haah 'fibonachi seudoprime' aahe. jevha "m" haah 500-bitcha motha aakada asato, tevha aapan matricks form vaparoon karyakshamatene "f" m "(modd" n ") moju shakato. tyamule
sootra _ 173
yethe matricks power "a" "m" hii modular ghatank vaparoon mojali jaate, jee matrixshee julavun ghetali jaau shakate.
fibonachi avibhajya.
fibonachi prime' hii eka fibonachi sankhya aahe jee prime aahe. pahilya kahee aaheth
hajaro ankansah fibonachi abhajya sankhya aadhalali aahe, parantu anant sankhya aahet qee nahee heey mahit nahee.
"f" "kn" "heey" "f" "n" "neey vibhajya aahe, mhanoon," "f" "4 = 3 vyatirikt, konatyahi fibonachi primecha prime index asane aavashyak aahe". sammishra sankhyanche aniyantritpane lamb dhavsankhya asalyane, sammishra fibonachi sankhyanche aniyantritpane lamb dhavsankhya dekhil aahet.
f" 6 = 8 peksha mothee konatihi fibonachi sankhya hii mulsankhyepeksha eka mothee kinwa eka qamee nasate.
144 hii ekmev kshullak nasaleli varg fibonachi sankhya aahe. 2001 madhye atila pethene siddha kele qee paripoorn shakti asalelya fibonachi sankhyanchi keval maryadit sankhya aahe. 2006 madhye, vaay. bugaud, emm. mignote aanee ess. 8 aanee 144 yaa asha keval kshullak nasalelya paripoorn shakti aahet heey sixekne siddha kele.
1, 3, 21 aanee 55 yaa ekmev trikoni fibonachi sankhya aahet, jyacha andaj vern hoggotne lavala hota aanee luo mingne siddha kelya hotya.
konatihi fibonachi sankhya paripoorn sankhya asoo shakat nahee. adhik samanyatah, 1 vyatirikt konatihi fibonachi sankhya gunakar paripoorn asoo shakat nahee aanee done fibonachi sankhyanche konatehi gunottar paripoorn asoo shakat nahee.
mukhya vibhajak.
1, 8 aanee 144 ("f" 1 = "f" 2 ", f" 6 aanee "f" 12) vagalata pratyek fibonachi sankhyecha eka avibhajya ghatak asato joou konatyahi lahan fibonachi sankhyecha ghatak nasato (karamayakalache pramey). parinami, 8 aanee 144 ("eff" 6 aanee "eff" 12) yaa ekmev fibonachi sankhya aahet jyaa itar fibonachi sankhyanche gunakar aahet.
fibonachi sankhyanchi avibhajya "pii" neey vibhajyata hii dantkathechya chinhachya sutrashi sambandhit aahe jyache moolyankan khalilpramane kele jaateh
sootra _ 175
jarr 'pii' hii eka abhajya sankhya asel tarr
sootra _ 176
udaharanarth,
sootra _ 177
assa avibhajya "pii" astitvat aahe qee nahee heey maheet nahee
sootra _ 178
asha avibhajyanna (jarr asateel tarr) bhint-surya-surya avibhajya mhatale jaeel.
tasech, jarr "p" ≤5 hii visham abhajya sankhya asel tarah
sootra _ 17
udaharan 1. "p" = 7, yaa prakaranat "p" shna 3 (modd 4) aanee aamchyakade aaheh
sootra _ 180
sootra _ 181
sootra _ 182
udaharan 2. "p" = 11, yaa prakaranat "p" shna 3 (modd 4) aanee aamchyakade aaheh
sootra _ 183
sootra _ 184
sootra _ 185
udaharan 3. "p" = 13, yaa prakaranat "p" shna 1 (modd 4) aanee aamchyakade aaheh
sootra _ 186
sootra _ 187
sootra _ 188
udaharan 4. "p" = 29, yaa prakaranat "p" shna 1 (modd 4) aanee aamchyakade aaheh
sootra _ 189
sootra _ 190
sootra _ 191
visham "n" sathee, "f" n "chey serve visham avibhajya vibhajak 1 modulo 4 chey ekroop aahet, yacha arth assa qee" f "n" chey serve visham vibhajak (visham avibhajya vibhajakanchi utpadane mhanoon) 1 modulo 4 chey ekroop aahet.
udaharanarth,
sootra _ 192
serve 'aay' <50000 sathee 'eff' ('aay' aanee 'hairsp') yaa fibonachi sankhyeche serve dnyat ghatak sambandhit bhandaranmadhye gola kele jaatat.
niyatkalikta modulo "aine".
jarr fibonachi anukramache sadasya modd "aine" ghetale gele, tarr parinami anukram jastit jaast 6 "aine" kalavadhisah niyatkalik asato. vividh 'aine' kalavadhichi lambi tathakathit pisano kalavadhi tayar karate. pisano kalavadhisathi samanya sootra nishchit karane hii eka khulee samasya aahe, jyamadhye upaprashni mhanoon modular purnankacha kinwa maryadit kshetratil ghatakacha gunakar kram shodhanyachya samasyeche eka vishesh udaharan samavisht aahe. tathapi, konatyahi vishisht 'aine' sathee, pisano kalavadhi heey chakr shodhanyache udaharan mhanoon aadhalu shakate.
samanyikaran.
fibonachi kram haah punaravrutti sambandhadvare aanee visheshtah resheey farak samikaranadvare paribhashit kelelya sarvat sopya aanee sarvat junya dnyat kramanpaiki eka aahe. yaa serve kramankade fibonachi kramache samanyikaran mhanoon pahile jaau shakate. visheshtah, sthir gunank asalelya sajatiy resheey farak samikaranache samadhan asalelya konatyahi kramakade binnetche sootra samanyikrut kele jaau shakate.
kahee vishisht udaharane jee kahee arthane fibonachi anukramachya jawal aahet, tyamadhye khaleel goshtincha samavesh hotoh
anuprayog.
ganeet.
pascalchya trikonachya "uthal" karnanmadhil dwipadi gunankanchi berries mhanoon fibonachi sankhya aadhalateh
sootra _ 193
heey nirmiti karya vadhavun siddha kele jaau shakate
sootra _ 194
aanee formula _ 195 chya atimpramane gola karane.
sootra kase vaparale jaate heey pahanyasathi, aapan upasthit sandnyanchya sankhyenusar berijchi vyavastha karoo shakatoh
jeey sootra _ 196 aahe, jithe aapan "n" − "k" − 1 padanmadhun "k" toojchi sthitee nivadat aahot.
yaa sankhya kahee mojmapavishayak samasyanvarhi upay detat, jyapaiki sarvat samanya mhanje dilelya sankhya 1s aanee 2s chee kramvarit berries mhanoon lihinyachya marganchi sankhya mojane (jyala rachana mhanatat); heey karnyache marg aahet (samatulya, heey _ 197 aayatakruti sootrachya dominojchi sankhya dekhil aahe). udaharanarth, eka velee eka kinwa done pavale uchalat 5 (3 chadhanyache marg aaheth
aakruti darshavate qee 8 chey 5 madhye vighatan kele jaau shakate (4 (3 chadhanyachya marganchi sankhya, tyanantar eka payari) adhik 3 (3 (3 chadhanyachya marganchi sankhya, tyanantar duheri payari). ekach ekach toch tark punaravrutti mhanoon lagoo keela jaato, jyapaiki chadhanyacha ekach marg aahe.
fibonachi sankhya binary stringchya sanchamadhye kinwa dilelya sanchatil upasanchanmadhye samaan ritine vegvegalya prakare aadhalu shakate.
nisarg.
zadanmadhye fandya padane, khodavar pananchi mandani, ananaschi fale, kelyachi phule yene aanee devdar shankuchi mandani aanee madhmashyanche kutumb vruksh yasarkhya jaivik paristhitinmadhye fibonachiche anukram disatat. keplarne nisargatil fibonachi anukramachi upasthiti darshavali, tyacha vapar kahee fulanche (suvarn gunottar-sambandhit) panchbhuj swaroop spasht karnyasathi keela. maidanachya manchavar bahutekda fibonachi sankhyanchya mojaneet pakalya asatat. 1830 madhye, keey. eff. skimper aanee aaa. bronla aadhalale qee vanaspatinche parastychi (sarpil phylotaxis) varanvar fibonachi sankhyancha samavesh asalele apurnank mhanoon vyakt kele jaatat.
vastavik udaharananna kahee pramanat mukt gatanvaril vishisht bijganitiy nirbandhanchi abhivyakti mhanoon, visheshtah vishisht lindenmeyer vyakaran mhanoon samajale jaau shakate, hii kalpana prazemislav prusinkiviz yanni pudhe aanali.
suryafulanchya mathyavaril fulanchya namunyasathi eka namuna 1979 madhye prastavit karnyat aala hota. yache swaroop asey aahe
sootra _ 199
fulancha nirdeshank kramank kuthe aahe aanee too eka sthir scaling ghatak aahe; asha prakare phuln farmetchya sarpilvar asatat. vichalan coone, andaje 137.51 °, haah suvarn coone aahe, joou vartulala suvarn gunottarat vibhajit karato. heey gunottar atarkik asalyamule, konatyahi fulacha shejari kendrapasoon agadi samaan konat nasato, tyamule fulanchi karyakshamatene pack hote. kaaran suvarn gunottarasathi tarksangat andaj heey swarupache asatat, tyamule kahee nirdeshankasathi floret kramankache sarvat jawalche shejari teey asatat, jeey kendrapasoonchya antaravar avalambun asatat. suryaful aanee tatsam fulanmadhye samanyatah ghadyalachya dishene aanee ghadyalachya viruddha dishene fulanche sarpil asatat, jeey samanyatah trijyechya sarvat bahereel shrenidware mojale jaatat.
khaleel niyamannusar, aadarsh madhmashyanchya vanshavalinmadhye dekhil fibonachi sankhya disun yeteh
asha prakare, narr madhmashyala nehmich eka palak asato aanee maadee madhmashyala done palak asatat. jarr ekhadyala konatyahi narr madhmashichi (1 madhmashi) vanshaval aadhalali, tarr tyala 1 palak (1 madhmashi), 2 aajee-aajoba, 3 panjoba, 5 panjoba ityadi aahet. palakanchya sankhyancha haah kram mhanje fibonachi kram. pratyek staravaril purvajanchi sankhya mhanje stree purvajanchi sankhya, mhanje, adhik purush purvajanchi sankhya, jee aahe. pratyek staravaril purvaj anyatha asambandhit aahet yaa avastav gruhitkakhali heey aahe.
heey lakshat aale aahe qee dilelya purvajanchya pidhitil manvi ex gunasutra varsa reshevaril sambhavya purvajanchi sankhya dekhil fibonachi kramache anusaran karate. purush vyaktila tyachya aaikadoon midalele ex gunasutra aanee tyachya vadilankadun midalele vaay gunasutra asate. purush tyachya swatachya ex gunasutrache (sootra _ 167) "mool" mhanoon ganala jaato aanee tyachya palakanchya pidhimadhye, tyache ex gunasutra ekach palkakadoon aale. purushachya aaeela tichya aaikadoon (mulachi aajee) eka ex gunasutra aanee tichya vadilankadun (mulache aajoba) eka gunasutra midale, mhanoon done aajee-aajobanche yogdan purush vanshajanchya ex gunasutrat hote. aaichya aajobanna tyanche ex gunasutra tyanchya aaikadoon midale aanee aaichya aajila tyanchya donhi palkankadun ex gunasutra midale, mhanoon teen aajee-aajobanche purush vanshajanchya ex gunasutrat yogdan hote. paach aajee-aajobanche purush vanshajanchya ex gunasutra ityadinmadhye yogdan hote. (heey asey gruheet dharate qee dilelya vanshatil serve purvaj swatantra aahet, parantu jarr konatihi vanshavali kalbahya jhaali tarr purvaj vanshavalichya anek olinvar disu lagatat, joparyant shevati vanshavalichya serve olinvar loksankhyecha sansthapak prakat hott nahee.)

