
yoonitti modulo enn-nte root

samkhyaa sidhaanthathil, positteev poornnasamkhyakalaaya "k", "n" ≤2-nuu k "yoonitti modulo" n-nte root, poornnasamkhyakalude valayathile yoonittiyude root aanu modulo "n"; athaayath, samavaakyathinte (allengil "kongruvans") formula _ 1-nulla pariharam "x". "x"-nuu "k" aanu ettavum cheria eksponant aanenkil, "x"-neey yoonitti modulo "n"-nte aadima "k"-aam root ennu vilikkunnu. notteshanum padaprayogatthinum modular ganithashaasthram kaanuka.
yoonitti moduloyude verukal krithyamaayi coprime aaya poornnasamkhyakalaanu. vaasthavathil, eee poornnasamkhyakal yoolarude sidhaantham anusarichu yoonitti moduloyude verukalaanu, mattu poornnasamkhyakal yoonitti moduloyude verukalakaan kazhiyilla, kaaranam avaa poojya bhajaka moduloyanu.
poornnasamkhyakalude moduloyude valayathinte yoonittukalude grouppinte janarettaraan premitteevu root modulo. formula _ 2, formula _ 3, formula _ 4 enniva yathaakramam carmichael function yoolarinte tottiyant function aanenkil maathrame premitteevu root modulo nilanilkkoo.
yoonitti moduloyude oru root formula _ 5-nte chila devicerukalkkulla yoonitti moduloyude aadimamaaya oru veraanu, neremarichu, yoonitti moduloyude aadimamaaya verukal formula _ 6-nte oru devisor aanenkil maathrame undaakoo.
aikyathinte verukal.
k" th verukalude ennam.
vyaapakamaayi amgeekarikkappetta chinnatthinte abhaavathil, yoonitti modulo "enn" nte "keey" verttukalude ennathe formula _ 11 upayogichu soochippikkunnu.
ithu niravadhi gunangale thripthippeduthunnu
udaaharanagal.
soothravaakyam _ 25 umm soothravaakyam _ 26 umm. eee saahacharyathil, aikyathin moonnu cube roottukal undu (1,2,4). ennirunnaalum, formula _ 27 aayirikkumbol, aikyathinte oru cube root maathrameyulloo, yoonit 1 thanne. poojyamallaatha oro samkhyaykkum "k" k "th roottukal ulla sankeernna samkhyakalude feeldil ninnu eee swabhaavam thikachum vyathyasthamaan.
aikyathinte aadima verukal.
aadima "k" th verukalude ennam.
vyaapakamaayi amgeekarikkappetta chinnatthinte abhaavathil, yoonitti modulo "enn" nte aadima "keey" verttukalude ennathe formula _ 40 kondu soochippikkunnu.
ithu thaazhepparayunna gunangale thripthippeduthunnu
x" ennathu yoonitti modulo "n"-nte aadima "k"-aamathe veraano ennu parishodhikkunnu.
phaast exponensiation vazhi, oraalkk aaa formula _ 1 parisodhikkaan kazhiyum. ithu shariyaanenkil, "x" ennathu yoonitti modulo "n" nte "k"-aam roottaanu, pakshe premitteevu aayirikkanamennilla. ithu oru aadima root allengil, soothravaakyam _ 63 ulla "k" yude chila devisor l undaakum. eee saadhyatha ozhivaakkunnathin, oru praimukond harikkappedunna "k" nuu thulyamaaya kurachu l-kall maathrame parishodhikkendathulloo. athaayath, parishodhikkendathu ithaanu
yoonitti modulo "enn"-nte oru aadima "keey"-aam root kandethuka.
aikyathinte aadima "k"-aamm verukalil, aadima soothravaakyam _ 29-aamm verukalaanu ettavum saadhaaranam.
athinaal oru aadima soothravaakyam _ 29th root aayirikkan chila poornnasamkhyakal pareekshikkaan shupaarsha cheiyunnu, athu vegathil vijayikk.
29-aamm root "x" enna aadima soothravaakyathinu, 36 enna samkhyayude soothravaakyam aikyathinte 31-aamm root enna aadima soothravaakyamaanu.
k" soothravaakyam _ 29 neey vibhajikkunnillenkil, aikyathinte "k"-aam verukal undaakilla.
onniladhikam aadima "k" th roottukal modulo "n" kandethunnu.
x" enna aikyathinte oru aadima "k"-aamm mooladhanam labhichukazhinjaal, oro shakti soothravaakyavum _ 71 ennathu aikyathinte 31-aamm moolasoothramaanu, pakshe athu oru aadimamaayirikkanamennil. formula _ 31 umm formula _ 76 umm coprime aanenkil maathram, power formula _ 71 aikyathinte 31-aamm root aanu. thelivu iprakaaramaanu formula _ 71 aadimamallengil, formula _ 80 ulla formula _ 31-nte oru divisar formula _ 78 nilavilundu, formula _ 31 umm formula _ 76 umm coprime aayathinal, formula _ 76 modulo formula _ 31-nte oru vipareeta formula _ 83 nilavilundu. ithu formula _ 86 nalkunnu, athaayath formula _ 35 oru aadima formula _ 31-aamm root allah, kaaranam cheria eksponant formula _ 78 undu.
athaayath, "x" enna padaprayogatthiloode oraalkk aikyathinte 90 vyathyastha aadima "k"-aamm verukal nedaanaakum, pakshe ivayellaam atharam verukalaayirikkilla. ennirunnaalum, avayellaam kandethunnathu athra eluppamalla.
yoonitti modulo "enn" nte aadima "keey" th root ulla oru "enn" kandethuka.
eathu poornnasankhya sheshikkunna klaas valayangalilaanu aikyathinte aadima "k" th root nilanilkkunnath? 31-dimensional intejar vector enna soothravaakyathinte oru vyathirikthamaya forrier parivartthanam (kooduthal krithyamaayi oru nambar saidhaanthika parivartthanam) kanakkakkaan ithu upayogikkaam. vipareeta parivartthanam nadatthaan, formula _ 31 kondu harikkuka; athaayath, "k" oru yoonit modulo formula _ 93 koodiyaanu.
atharam oru "n" kandethunnathinulla oru lalithamaaya maarggam, ganitha purogathi soothravaakyathile moduliyumaayi bandhappett aadima "k" th roottukal parisodhikkuka ennathaanu-94 eee modulikalellaam "k"-nuu coprime aanu, athinaal "k" oru yoonittaan. ganitha purogathikalekkurichulla diriklettinte sidhaantham anusarichu purogathiyil ananthamaaya niravadhi primukal undu, oru prime formulakku _ 20, athu formula _ 96 kaivasham vaykkunnu. athinaal formula _ 97 prime aanenkil, formula _ 98, athinaal aikyathinte aadima "k" th verukal undu. ennaal primukalkkulla parisodhana valare shakthamaan, koodaathe mattu uchithamaaya moduli undayirikkaam.
yoonitti modulo "enn"-nte onniladhikam aadima verukalulla oru "enn" kandethuka.
yoonitti modulo formula 28-nte 100 roottukal ulla oru modulus formula _ 28 kandethunnathinu, inipparayunna sidhaantham prashnathe lalithamaaya onnilekku kuraykkunnu
pinnottulla disha
yoonitti modulo formula 28-nte 105-aamm root enna oru aadima formula formula 35 aanenkil, formula 109 ennathu yoonitti modulo formula 28-nte 110-aamm root formulayanu.
munnottulla disha
yoonitti modulo formula _ 28-nte aadima soothravaakyam _ 103 roottukal undengil, ellah eksponant formula _ 114 umm formula _ 29-nte devicerukalaanu. ithu formula _ 116 soochippikkunnu, ithinartham yoonitti modulo formula _ 28-nte oru aadima soothravaakyam _ 105-aamm root undu ennaanu.

