
tonelli-shanks algoridam

tonelli-shanks algoridam (shanksdware resol algoridam mhanoon sandarbhit) haah modular ankganitacha vapar "r" sathee "r" 2 shna "n" (modd "p") yaa swarupat sarvasamaveshaktemadhye sodavanyasathi keela jaato, jethe "p" haah avibhajya aaheh mhanje, "n" modulo "p" chey vargamool shodhanyasathi.
tonelli-shankscha vapar sammishra modulinsathi keela jaau shakat naheeh vargamool modulo sammishra sankhya shodhane hii purnank ghatakanashi samatulya eka sanganakeey samasya aahe.
yaa algoridamchi samatulya, parantu thodi adhik anavashyak aavrutti vikasit kelee gelee
1891 madhye. yethe charcha keleli aavrutti 1973 madhye daniele shanksney swatantrapane vikasit kelee hotee, jyane spasht keleh
yaa aitihasik sandarbhambaddal jaanun ghenyat mallah vilamb jhaala kaaran mea eka mitrala dicksonchya itihasacha pahila khand deela hota aanee too kadhihi parat keela gela nahee.
dixonchya matey, tonelicha algoridam avibhajyanshanshivay 'x' modulo prime power 'pλ' chey vargamoole gheu shakato.
moolbhut kalpana.
shoonya nasalele sootra _ 1 aanee visham avibhajya sootra _ 2 dile tarr, yoolarcha nikash aaplyala sangato qee sootra _ 1 chey vargamool (i.eae.sootra _ 1 heey eka dvighat avashesh aahe) jarr aanee fakt jarah
yaulat, jarr _ 6 yaa sankhyechya sutrala vargamool nasel (avashesh nasalela), tarr yularcha nikash aaplyala sangato keeh
asey sootra _ 6 shodhane katheen nahee, kaaran 1 aanee sootra _ 9 madheel ardhya purnankanmadhye haah gunadharm aahe. mhanoon aapan asey gruheet dharato qee aaplyala asha avasheshanmadhye pravesh aahe.
(samanyatah) 2 neey varanvar bhag dilyane, aapan sootra _ 9 heey sootra _ 11 mhanoon lihu shakato, jethe sootra _ 12 visham aahe. lakshat ghya qee jarr aapan prayatna keela tarr
mug sootra 14. jarr sootra 15 asel tarr sootra 16 heey sootra 1 chey vargamool aahe. anyatha, sootra 18 sathee, aaplyakade sootra 16 aanee sootra 20 samadhankarak aaheh
jarr, vareel sutranche samadhan karnare vishisht sootra _ 27 sathee sootra _ 16 aanee sootra _ 20 chaa paryay deela (jethe sootra _ 16 heey sootra _ 1 chey vargamool nahee), tarr aapan sootra _ 32 sathee aanakhi eka sootra _ 16 aanee sootra _ 20 sahajpane moju shakato jenekarun vareel sambandh tikoon rahtil, tarr sootra _ 20 heey 1, i chey sootra _ 34 weey mool hoiparyant aapan yachee punaravrutti karoo shakato.eae.sootra _ 35. tyaa velee sootra _ 16 heey sootra _ 1 chey vargamool aahe.
sootra _ 20 heey sootra _ 39 chey mool aahe qee nahee heey aapan tyache sootra _ 40 vela varg karoon tapasu shakato aanee teey 1 aahe qee nahee heey tapasu shakato. jarr asel tarr aaplyala kahihi karanyachi aavashyakta nahee, sootra _ 16 aanee sootra _ 20 chee samaan nivad karya karate. parantu jarr tase nasel tarr sootra _ 43 heey-1 asane aavashyak aahe (kaaran varg kelyane teey 1 dete aanee 1 modulo sootra _ 2 chey fakt done vargamoole 1 aanee-1 asoo shakatat).
sootra _ 16 aanee sootra _ 20 chee naveen jodee shodhanyasathi, aapan sootra _ 16 laa ghatak sootra _ 48 neey gunakar karoo shakato, jeey nirdharit kele jaeel. mug sootra _ 21 thevanyasathi sootra _ 20 laa ghatak sootra _ 50 neey gunakar karane aavashyak aahe. mhanoon aaplyala ghatak sootra _ 50 shodhane aavashyak aahe jenekarun sootra _ 53 heey 1 chey sootra _ 39-weey mool aahe, kinwa samatulya sootra _ 50 heey-1 chey sootra _ 39-weey mool aahe.
yethe yukti mhanje dnyat avashesh nasalele sootra _ 6 vaparane. varr darshavilelya sootra _ 6 varr lagoo kelela yoolarcha nikash mhanato qee sootra _ 59 heey-1 chey sootra _ 60 weey mool aahe. mhanoon sootra _ 59 chaa varanvar varg karoon, aaplyala sootra _ 62-weey mool-1 chya anukramat pravesh aahe. aapan sootra _ 48 mhanoon kaam karnyasathi yogya eka nivadu shakato. thode badalanari dekhbhal aanee kshullak cases sumpedens, khaleel algoridam naisargikaritya udbhavato.
algoridam.
purnankanchya gunakar gatachya ghatkanvaril karye aanee tulana modulo pii sootra _ 64 heey apratyakshapane modd "pii" aahet.
inputah
utpadaneh
algoridamah
ekada tumhi 'r' saha sarvasamaveshakata sodavali qee dusare uttar sootra _ 73 aahe. jarr kiman 'i' asey sootra _ 70 'm' asel tarr sarvasamaveshakatache konatehi uttar astitvat nahee, i.eae. "aine" haah dvighat avashesh nahee.
jevha "pii" shna 1 (modd 4) asel tevha heey sarvat upayukt tharate.
p" shna 3 (modd 4) asha avibhajyansathi, yaa samasyeche sambhavya nirakaran sootra _ 75 aahe. jarr heey sootra _ 76 purn karatat, tarr teey ekmev upay aahet. naslyas, sootra _ 77, "aine" heey eka dvighat avashesh nasalele aahe aanee tyavar konatehi upay naheet.
purava.
loopchya pratyek punaravruttichya suruvatila khaleel loop aparivartaniy asatat heey aapan dakhavu shakatoh
suruvatilah
pratyek punaravruttivar, 'm', 'c', 't', 'r' saha, 'm', 'c', 't', 'r' chya jaagi naveen moolye yetatah
sootra 79 aanee loopchya suruvatila 'tii' = 1 viruddha chachani, aapan pahato qee aaplyala nehmi 0 <'aay' <'emm' madhye 'aay' asey aadhalel qee sootra _ 70. emm 'pratyek punaravruttivar katekorpane lahan aahe aanee asha prakare algoridam thambanyachi hummy aahe. jevha aapan 'tii' = 1 hii aut thokato aanee thambato, tevha shevatchya loopcha aparivartaniy assa arth hoto qee 'aar' 2 = 'aine'.
tii" chaa kram.
ghatakanchya kramacha vapar karoon aapan loop aparivartaniy goshti paryayipane vyakt karoo shakatoh
algoridamchi pratyek payari 'tii' chaa achook kram mojoon aanee tyach kramachya ghatakane gunakar karoon 'tii' laa eka lahan upasamuhat halavate.
udaharanadakhal.
r" 2 shna 5 (modd 41) hii susangatata sodavane. aavashyakatenusar 41 haah avibhajya aahe aanee 41 shna 1 (modd 4) aahe. 5 heey yularchya nikashannusar eka dvighat avashesh aaheh sootra _ 97 (purvipramane, sootra _ 98 madheel karye apratyakshapane modd 41 aahet).
kharey tarr, 282 shna 5 (modd 41) aanee (−28) 2 shna 132 shna 5 (modd 41). tyamule algoridam aaplya susangatatesathi done upay dete.
algoridamcha veg.
tonelli-shanks algoridamsathi aavashyak aahe (serve sambhavya input (vargatmak avashesh aanee vargatmak nasalele avashesh))
modular gunakar, jethe sootra 129 heey sootra _ 2 chya dwaiti pratinidhitvatil ankanchi sankhya aahe aanee sootra _ 131 heey sootra _ 2 chya dwaiti pratinidhitvatil ankanchi sankhya aahe. jarr aavashyak chaturbhuj avashesh sootra _ 6 yadruchchikpane ghetalele sootra _ 134 heey eka chaturbhuj avashesh aahe qee nahee heey tapasoon shodhayache asel, tarr tyasathi (sarasarivar) sootra _ 135 hii dantkathechya chinhachi ganana aavashyak aahe. dantkathechya chinhachya done ganananchi sarasari khalilpramane spasht kelee aaheh sootra 134 heey aakasmik sootra _ 137 asalele eka dvighat avashesh aahe, jeey sootra _ 138 peksha lahan aahe parantu sootra _ 139 aahe, mhanoon aaplyala sootra _ 134 heey dvighat avashesh aahe qee nahee heey sarasari tapasanyachi aavashyakta aahe.
heey moolatah darshavite qee jarr modulus sootra _ 2 yadruchchik asel tarr tonelli-shanks algoridam khoop changle karya karate, mhanje jarr sootra _ 2 chya dvisangi pratinidhitvatil ankanchya sankhyechya sandarbhat sootra _ 142 visheshtah mothe nasel tarr varr lihilyanusar, sipollacha algoridam tonelli-shankspeksha changle karya karato jarr (aanee fakt jarr) sootra _ 144 asel.
tathapi, jarr tyaeivaji sootra _ 145 chya 2-sayalo upasamuhat swatantra laghuganak ganana karnyasathi s′ chaa algoridam vaparla tarr sootra _ 147 neey asymptotically baddh asalelya abhivyaktine sootra _ 145 badaloo shakato. spashtapane, sootra _ 148 ashee ganana kelee jaate qee sootra _ 149 aanee nantar sootra _ 150 sootra _ 151 chey samadhan karate (lakshat ghya qee sootra _ 148 heey 2 chey gunakar aahe kaaran sootra _ 1 heey eka chaturbhuj avashesh aahe).
algoridamsathi aaplyala eka chaturbhuj avashesh sootra _ 6 shodhane aavashyak aahe. asey sootra _ 6 shodhanyasathi bahupadi velet chalanare konatehi dnyat nirdharak algoridam nahee. tathapi, jarr samanyikrut reeman gruhitak kharey asel tarr eka chaturbhuj avashesh sootra _ 156 astitvat aahe, jyamule tyaa maryadeparyantche pratyek sootra _ 6 tapasane aanee bahupadi velechya aat yogya sootra _ 6 shodhane shakya hote. tathapi, lakshat thewa qee hii eka sarvat white paristhiti aahe; sarvasadharanpane, varr namood kelyapramane sarasari 2 chachanyanmadhye sootra _ 6 aadhalate.
vapar.
jyaa konatyahi prakriyet vargamool modulo prime aavashyak aahe, tyasathi tonelli-shanks algoridam (naisargikaritya) vaparla jaau shakato. udaharanarth, lambvartulakar vakrareshevaril bindu shodhanyasathi yacha vapar keela jaau shakato. heey rabin guptapranalitil gananesathi aanee dvighat chalanichya chhananichya tappyasathi dekhil upayukt aahe.
samanyikaran.
tonelli-shanksche samanyikaran konatyahi chakriy gatamadhye (sootra _ 160 aivaji) aanee aniyantrit purnank "keey" sathee "keey" vya mulanmadhye kele jaau shakate, visheshtah maryadit kshetrachya ghatakache "keey" weey mool ghenyasathi.
jarr anek chauras-mule ekach chakriy gatat kelee gelee pahijet aanee s khoop mothe nasel, tarr 2-shakti kramachya ghatakanchya chauras-mulanchi sarani aagau tayar kelee jaau shakate aanee algoridam khalilpramane sope aanee vegwan keela jaau shakato.
tonelicha algoridam modd pii. keey. varr kaam karel.
dixonchya 'sankhyanchya siddhantanusar"
aaa. tonelline _ 166 yaa sootrachya mulansathi eka spasht sootra dile
dickson sandarbh sootra _ 167 chya vargmoolasathi khaleel sootra darshavito.
teey sootra _ 175 lakshat gheuun aanee teey sootra _ 176 lakshat gheuun
aanakhi eka udaharan gheuyah sootra _ 178 aanee
dixon dekhil khaleel samikaranala toneliche shrey detoh
183 sootra vaparoon aanee 184 sutrache mapank vaparoon ganeet khalilpramane aaheh
pratham, modular vargamool modd sootra _ 2 shodha jeey niyamit tonelli algoridamdware kele jaau shakateh
aanee toneliche samikaran lagoo karane (varr pahaa):
dixoncha sandarbh spashtapane darshavito qee tonelicha algoridam _ 190 yaa sootrachya modulinvar karya karato.