
punha qaraa

manoranjak ganitat, punaravrutti hii 11,111 kinwa 1111 sarkhi sankhya aahe jyamadhye fakt 1 haah ank asato-adhik vishisht prakarcha punarganiti. haah shabd punaravrutti ekkasathi aahe aanee 1966 madhye albert etch yanni tayar keela hota. recreation inn daad theary auph numbers' yaa tyanchya pustakat bealer.
repunit prime hii eka repunit aahe jee eka prime number dekhil aahe. bases-2 madheel repunits asalelya avibhajya ghatak heey mersen avibhajya ghatak asatat. meey 2023 paryant, sarvat mothee dnyat abhajya sankhya, sarvat mothee sambhavya abhajya "r" 8177207 aanee sarvat mothee lambvartulakar vakr prathamikata-siddha abhajya "r" 86453 hii serve vividh payanmadhye punaravartit aahet.
vyakhya.
bases-'buub' repunitschee vyakhya ashee kelee jaate (heey 'buub' ektar sakaratmak kinwa nakaratmak asoo shakate)
asha prakare, "r" "n" ("b") yaa aakadyamadhye aadhar-"b" pratinidhitvatil 1 ankachya "n" pratee asatat. pahilya done punarbhujyancha aadhar-'aine' = 1 sathee 'bii' aanee 'aine' = 2 heey aahet 
visheshtah, 'dashansh (aadhar-' 10 ') punarutpadan' jyanna anekada fakt 'punarutpadan' mhanoon sambodhale jaate tyanchi vyakhya ashee kelee jaateh
asha prakare, "r" "n" = "r" "n" (10) yaa aakadyat 10chya aadharatil 1 ankachya "n" pratee asatat. bases-10 yaa punarutpadanacha kram yapasoon suru hoto
tyachpramane, bases-2 yaa repunitschee vyakhya ashee kelee jaateh
asha prakare, "r" "n" (2) yaa sankhyemadhye aadhar-2chya pratinidhitvatil 1 ankachya "n" pratee asatat. kharey tarr, bases-2 repunits hii suprasiddha mersen sankhya "m" "n" = 2 "n"-1 aahet, jyapasoon teey suru hotat.
dashansh pratikrutinche gunankan.
(mukhya ghatak rangeet mhanje "naveen ghatak", i. eae. mukhya ghatak 'r' 'n' laa vibhajit karato parantu serve 'k' <'n' sathee 'r' k 'laa vibhajit karat nahee) 
n"> 1 sathee "r" n "chaa sarvat lahan abhajya ghatak aaheh
avibhajya punarbharan qaraa.
asha sankhyanche pramukh ghatak shodhat asalelya manoranjak ganitshastradnyanni punaruktichi vyakhya prerit kelee hotee.
jarr 'aine' haah 'aii' neey vibhajya asel tarr 'aar' aine '(' bii ') haah' aar 'aii' ('bii') neey vibhajya aahe heey dakhavane sope aaheh
jethe sootra _ 6 heey sootra _ 7 cyclotomic bahupadi aahe aanee "n" chya vibhajkanvar "d" shreni aahe. pii" primesathi,
jevha "x" laa "b" neey badalale jaate tevha tyache repunitche apekshit swaroop asate.
udaharanarth, 9 heey 3 neey vibhajya aahe aanee asha prakare "r" 9 heey "r" 3 neey vibhajya aahe-kharey tarr, 111111111 = 111·1001001. sambandhit chakriy bahupadi sootra _ 9 aanee sootra _ 10 heey anukrame sootra _ 11 aanee sootra _ 12 aahet. tyamule, "r" "n" haah avibhajya honyasathi, "n" haah avibhajya asane aavashyak aahe, parantu "n" haah avibhajya asane purese nahee. udaharanarth, "r" 3 = 111 = 3·37 haah avibhajya nahee. r" 3 "chey heey prakaran vagalata, jarr kahee" k "sathee" p "= 2" kn "+ 1 asel tarr" n "sathee" r "" n "laa fakt vibhajit karoo shakato.
dashansh punarbhutik avibhajya.
"r" "n" "haah" n "= 2,9,23,317,1031,49081,86453 sathee avibhajya aahe"... (oo. eae. aay. ess. madheel kram). 3 april 2007 roji harvey doubnerne (jyala 'aar' 49081 dekhil aadhalale) jaheer kele qee 'aar' 109297 haah sambhavya avibhajya aahe. 15 julai 2007 roji maxim vozanine 'aar' 270343 haah bahudha avibhajya asel ashee ghoshana kelee. serge batalov aanee rayan proper yanna anukrame 20 april aanee 8 meey 2021 roji "aar" 5794777 aanee "aar" 8177207 heey sambhavya avibhajya asalyache aadhalale. tyanchya shodhanusar, pratyek dnyat sambhavya avibhajya sarvat motha hota. 22 march 2022 roji sambhavya avibhajya "aar" 49081 akheris avibhajya asalyache siddha jhaale. 15 meey 2023 roji sambhavya abhajya "r" 86453 akheris abhajya asalyache siddha jhaale.
assa andaj lavala gela aahe qee anant repunit prime aahet aanee teey sadharanpane prime number prameyane andajit kelyapramane ghadatat asey disateh "n" th repunit primecha ghatank samanyatah ("n" −1) th chya ghatkachya nishchit gunakajaval asato.
mukhya pratikruti haah kramvardhit avibhajyancha eka kshullak upasanch aahe, i.eae., jeey avibhajya tyanchya ankanchya konatyahi kramparivartanantar avibhajya rahatat.
vishisht gunadharm aahet
bases 2 repunit prime.
bases-2 repunit primena mersen prime mhanatat.
bases 3 repunit prime.
pahilya kahee bases-3 repunit avibhajya aahet 
sootra _ 13 shhee sambandhit 
aadhar 4 punarbhujyaman avibhajya.
ekmev bases-4 repunit prime 5 (sootra _ 14) aahe. sootra _ 15 aanee 3 nehmi sootra _ 16 vibhajit karatat jevha "n" visham asate aanee sootra _ 17 jevha "n" summ asate. 2 peksha mothe 'n' sathee, sootra _ 16 aanee sootra _ 17 donhi 3 peksha mothe aahet, mhanoon 3 chaa ghatak kadhoon takalyane ajunahi 1 peksha mothe done ghatak rahatat. mhanoon, sankhya avibhajya asoo shakat nahee.
bases 5 repunit prime.
pahilya kahee bases-5 repunit avibhajya aahet
sootra _ 13 shhee sambandhit
aadhar 6 punarbharan avibhajya.
pahilya kahee bases-6 repunit avibhajya aahet
sootra _ 13 shhee sambandhit
bases 7 repunit prime.
pahilya kahee bases-7 repunit prime aahet
sootra _ 13 shhee sambandhit
aadhar 8 punarbharan avibhajya.
ekmev bases-8 repunit prime 73 aahe (sootra _ 23). sootra _ 24 aanee 7 heey sootra _ 25 laa vibhajit karatat jevha "n" 3 neey vibhajya nasate aanee sootra _ 17 jevha "n" haah 3 chaa gunakar asato.
aadhar 9 punarbhujyaman avibhajya.
bases-9 repunit prime naheet. sootra _ 27, aanee sootra _ 28 aanee sootra _ 29 donhi summ aanee 4 peksha mothe aahet.
aadhar 11 punarbhujyaman avibhajya.
pahilya kahee bases-11 repunit prime aahet
sootra _ 13 shhee sambandhit
aadhar 12 punarbharan avibhajya.
pahilya kahee bases-12 repunit avibhajya aahet
sootra _ 13 shhee sambandhit
bases 20 repunit prime.
pahilya kahee bases-20 repunit avibhajya aahet
sootra _ 13 shhee sambandhit
bases "b" asey qee "rp" (b) haah prime "p" sathee prime aahe.
sarvat lahan aadhar sootra _ 33 mhanje sootra _ 34 heey avibhajya aahe (jethe sootra _ 35 heey sootra _ 13 weey avibhajya aahe)
sarvat lahan aadhar sootra _ 33 mhanje sootra _ 38 heey avibhajya aahe (jethe sootra _ 35 heey sootra _ 13 weey avibhajya aahe)
punarbharan abhajya moolank aadhar "buub" chee yadee.
sarvat lahan abhajya sootra _ 41 mhanje sootra _ 34 heey abhajya aahe (sootra _ 43 pasoon prarambh qaraa, jarr asey konatehi sootra _ 35 astitvat nasel tarr 0)
sarvat lahan abhajya sootra _ 41, jase qee sootra _ 38 heey abhajya aahe (sootra _ 43 pasoon prarambh qaraa, jarr asey konatehi sootra _ 35 astitvat nasel tarr, hii sandnya sadhya adnyat asalyas prashnachinh)
adhik mahitisathi, pahaa.
samanyikrut punarbhutik sankhyanche bijganit ghatakakaran.
jarr 'buub' hii eka paripoorn shakti asel ('emm' aine 'mhanoon lihili jaau shakate,' emm 'saha,' aine 'purnank,' aine '> 1) tarr 1 pasoon bhinn asel, tarr bases-' bii 'madhye jastit jaast eka repunit asel. jarr "n" hii eka abhajya shakti asel ("p" r "mhanoon lihile jaau shakate," p "abhajya" saha, r "purnank" p "," r "> 0), tarr aadhar-b madheel serve punarbhugikarane" rp "aanee" r2 "vyatirikt abhajya naheet. "aar. pii." heey ektar avibhajya kinwa sammishra asoo shakate, purvichi udaharane, "bii" = −216, −128,4,8,16,27,36,100,128,256, ityadi.nantarchi udaharane, "b" = −243, −125, −64, −32, −27, −8,9,25,32,49,81,121,125,144,169,196,216,225,243,289, ityadi., aanee "r2" haah avibhajya asoo shakato (jevha "p" 2 pasoon vegala asato) keval jarr "b" haah nakaratmak asel, tarr −2 chee ghaat, udaharanarth, "b" = −8, −32, −128, −8192, ityadi., kharey tarr, "r2" heey sammishra dekhil asoo shakate, udaharanarth, "b" = −512, −2048, −32768, ii. jarr "n" hii abhajya shakti nasel, tarr konatahi aadhar-"b" punarutpadak abhajya astitvat nahee, udaharanarth, "b" = 64,729 ("n" = 6 saha), "b" = 1024 ("n" = 10 saha), aanee "b" = −1 kinwa 0 (konatyahi naisargik sankhyes). dusari vishesh paristhiti mhanje "b" = −4 "k" 4, jyamadhye "k" sakaratmak purnank aahe, jyamadhye aurifulian factorization aahe, udaharanarth, "b" = −4 ("k" = 1 saha, nantar "r2" aanee "r3" heey avibhajya aahet), aanee "b" = −64, −324, −1024, − 2500, −5184,... ('keey' = 2,3,4,5,6 saha...), tarr konatahi bases-"buub" repunit prime astitvat nahee. asehi manale jaate qee jevha "b" hii paripoorn shakti nasate kinwa "k" sakaratmak purnankasah −4 "k" 4 nasate, tevha anant anek bases-"b" repunit prime asatat.
samanyikrut punarbharan anuman.
samanyikrut repunit avibhajyanshi sambandhit eka anumanah (pudhil samanyikrut mersen avibhajya kuthe aahe yacha andaj, jarr andaj kharaa asel tarr serve bases sutransathi anant anek repunit avibhajya aahet.
konatyahi purnank sutrasathi _ 33, jeey atty purn karateh
formchya samanyikrut repunit prime aahet
abhajya sootra _ 35 sathee, abhajya sankhya sarvottam julananya reshejaval vitarit kelya jaatil.
jethe sootra _ 65, sootra _ 66 maryada aahe
aanee tyabaddal
bases-"buub" heey "aine" peksha qamee moolank punarbhugikrut karatat.
aamchyakade khaleel 3 gunadharm dekhil aaheth
itihas.
tyaa velee tyaa navane olakhale jaat nasale taree, ekonisavya shatkat anek ganitshastradnyanni punaravrutti dashanshanchya chakriy namunyancha abhyas aanee andaj lavnyachya prayatnat bases-10 madheel repunitscha abhyas keela.
heey agadi lavkar aadhalun aale hote qee 5 peksha jaast asalelya konatyahi avibhajya "pii" sathee, 1/"pii" chya dashansh vistaracha kalavadhi haah "pii" neey vibhajya asalelya sarvat lahan punarbhukt sankhyechya lambichya barobaricha asato. 60, 000 paryantchya avibhajyanchya paraspar kalavadhichi saranya 1860 paryant prakashit karnyat aalee hotee aanee asha ganitshastradnyanni "aar 16" paryantchya serve repunitsche aanee anek mothya repunitsche punha punarujjivan mhanoon factorization karnyas parvanagi dilee hotee. 1880 paryant, agadi "aar 17" teey "aar 36" dekhil ghatak kele gele hote aanee heey kutuhalpurna aahe qee, edward lukasne jaree tees lakhanpeksha qamee kalavadhicha ekonis kalavadhi nasalela prime darshavila asala taree visavya shatkachya suruvatiparyant prathmikatesathi konatyahi punaruktichi chachani ghenyacha prayatna keela gela navhata. american ganitshastradnya auskar hoppey yanni 1916 madhye 'aar19' haah avibhajya asalyache siddha kele aanee lehmer aanee cratchick yanni swatantrapane 1929 madhye 'aar23' haah avibhajya asalyache shodhun kadhale.
1960 chya dashkaparyant, jevha sangankandware punarutpadanache anek naveen ghatak shodhale jaau lagale aanee mukhya kalavadhichya aadhichya sarnyanmadhil antar durust kele gele, tevha punarutpadanachya abhyasat aanakhi pragati jhaali nahee. "r317" haah sumare 1966 saalee sambhavya avibhajya asalyache aadhalun aale aanee akra varshannantar jevha "r1031" haah daha hajaranpeksha qamee ankansah aanakhi sambhavya avibhajya punarbhutik asalyache darshavile gele tevha too avibhajya siddha jhaala. 1986 madhye too sarvoch siddha jhaala, parantu pudhil dashakat pudhil pramukh repunitsasathiche shodh satatyane ayashasvi jhaale. tathapi, samanyikrut repunitschya kshetrat eka motha side-development hota, jyamule mothya sankhyene naveen prime aanee sambhavya prime tayar jhaale.
1999 pasoon, aanakhi chaar bahudha pramukh repunits sapadale aahet, parantu tyanchya prachand aakaramule najikchya bhavishyat tyapaiki konatehi pramukh siddha honyachi shakyata nahee.
cunningham prakalp 2,3,5,6,7,10,11 aanee 12 yaa aadharavar (itar sankhyambarobarach) gunankanchya purnank ghatakanche dastaivajikaran karnyacha prayatna karato.
demlo sankhya.
dii. aar. kaprekar yanni demlo sankhyanchi vyakhya davya, madhya aanee ujavya bhagache sanyojan ashee kelee aahe, jithe dawa aanee ujava bhag samaan lambicha (davikade sambhavya agraganya shunyaparyant) asava aanee punha ankee sankhyeshi jodala pahije aanee madhalya bhagat yaa punaravrutti kelelya ankachi konatihi atirikt sankhya asoo shakate. 
tyanna tatkalin jee. varr bombepasoon 30 mail antaravar asalelya demlo railway sthanakapasun (aata jyala dombivili mhanatat) naav denyat aale aahe.aay.pii. railway, jithe kaprekar yanni tyancha tapas suru keela.
too 1,121,12321,1234321 yaa formchya 'adbhut demlo numbers' mhanato..., 12345678987654321. heey repunitsche varg aahet yaa vastusthitimule kahee lekhakanni demlo sankhyanna hyancha 1,121,12321, haah anant kram mhatale aahe..., 12345678987654321,1234567900987654321,123456790120987654321,..., jaree heey "pii" = 10,19,28, sathee demlo sankhya naheet heey tapasale jaau shakate...