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aMkelu-saMkhyalu: rAmAnujan nuMDi bhArgava dAkA

dArina pOyE dAnayyani Api ‘ayin^sTayin EmiTi chEsEDayA?’ ani aDigitE mUDoMtulu saraina samAdhAnamE rAvachchu. kAnI, rAmAnujan EmiTi chEsEDaMTE -okka TEksI kathani minahAyiMchi – sAmAnyulu evvarU sari ayina samAdhAnaM cheppalEru.

rAmAnujan aMkelatO chEsina anEkamaina gAraDIlalO oka dAnini nalugurikI ardhaM ayE rItilO cheppaTAniki prayatnistAnu. muMdastugA 1, 4, 9, 16, 25 36, modalaina saMkhyalatO katha modalu peDatAnu. E unnata pAThaSAla vidyArdhi ayinA sarE I saMkhyalalO bANIni iTTE pasigaTTa galadu. vITini varga saMkhyalu (#square numbers)# aMdAM. eMdukaMTE ivi 1, 2, 3, 4, 5, 6, modalaina saMkhyalani vargIkariMchagA (aMTE, oka saMkhyani dAni tOTE guNiMchaTaM) vachchina saMkhyalu kanuka. vITinE konni saMdarbhAlalO chadarapu saMkhyalu ani kUDA anaTaM kaddu. I varga saMkhyalaki unna pratyEkata EmiTO chinna udAharaNa dvArA vivaristAnu. mIku tOchina pUrNa saMkhya (#integer)# ni tIsukoMDi. I pUrNa saMkhyani konni varga saMkhyala mottaMgA rAyochchu. udAharaNaki: 10 = 1 + 1 + 4 + 4. maroka udAharaNa: 30 = 1 + 4 + 9 + 16.

sA. Sa. 1770 lO phrAnsu dESapu gaNita SAstravEtta jOseph luyI lagrAMj oka siddhAMtAnnirujuvu chESAru: prati dhana pUrNa saMkhya (#positive integer)# tanaMta tAnugA oka varga saMkhya ayinA ayi uMDAli, lEdA reMDu kAni, mUDu kAni, nAlugu kAni varga saMkhyala mottamayinA ayi uMDAli. eTTi paristhitulalOnU nAlugu vargala mottaM (#x2 + y2 + z2 + t2)# miMchi avasaraM uMDadu.

lagrAMj rujuvu chEsina sUtraM lO unna (#x2 + y2 + z2 + t2)# vaMTi gaNita rUpAnni varga rUpaM (#quadratic form)# aMTAru. I varga rUpAla svabhAvaM ardhaM ayina veMbaDi, dhana pUrNa saMkhyalani abhivarNiMchaTAniki iTuvaMTi varga rUpaM idokkaTEnA lEka iMkA unnAyA ani anumAnaM rAnE vachchiMdi. rAvaTaM aMTE vachchiMdi kAni I samasyaki parishkAraM uMdO lEdO okaTinnara SatAbdAla varakU evvarikI teliyalEdu.

iMtalO, 1916 lO, SrInivAsa rAmAnujan “idokkaTE kAdu. iTuvaMTi varga rUpAlu mottaM 53 unnAyi” ani vATi jAbitA rAsi ichchEsEDu! udAharaNaki prati saMkhyani oka varga, reMDiMtala varga, mUDiMtala varga, nAlugiMtala vargAla mottaM (1.#x2 + 2.y2 + 3.z2 + 4.t2)# gA rAyavachchannAru Ayana. kutUhalaMtO kutakuta lADE prANulaki I 53 rUpAlU I diguva paTTikalO chUpeDatAnu. I paTTikalO vADina sUtraM (#a.x2 + b.y2 + c.z2 + d.t2)# anukuMTE iMdulO #a, b, c, d# la viluvalu elA uMTAyO varusagA chUpiMchEnu.

#[1, 1, 1, 2], [1, 1, 1, 3], [1, 1, 1, 4], [1, 1, 1, 5], [1, 1, 1, 6], [1, 1, 1, 7], [1, 1, 2, 3], [1, 1, 2, 3], [1, 1, 2, 4], [1, 1, 2, 5], [1, 1, 2, 6], [1, 1, 2, 7], [1, 1, 2, 8], [1, 1, 2, 9], [1, 1, 2, 10], [1, 1, 2, 11], [1, 1, 2, 12], [1, 1, 2, 13], [1, 1, 2, 14], [1, 1, 3, 3], [1, 1, 3, 4], [1, 1, 3, 5], [1, 1, 3, 6], [1, 2, 2, 2], [1, 2, 2, 3], [1, 2, 2, 4], [1, 2, 2, 5], [1, 2, 2, 6], [1, 2, 2, 7], [1, 2, 3, 3], [1, 2, 3, 4], [1, 2, 3, 5], [1, 2, 3, 6], [1, 2, 3, 7], [1, 2, 3, 8], [1, 2, 3, 9], [1, 2, 3, 10], [1, 2, 4, 4], [1, 2, 4, 5], [1, 2, 4, 6], [1, 2, 4, 7], [1, 2, 4, 8], [1, 2, 4, 9], [1, 2, 4, 10], [1, 2, 4, 11], [1, 2, 4, 12], [1, 2, 4, 13], [1, 2, 4, 14], [1, 2, 5, 6], [1, 2, 5, 7], [1, 2, 5, 8], [1, 2, 5, 9], [1, 2, 5, 10].

#rAmAnujan sAdhiMchina phalitaM avagAhana kAgAnE gaNitakulaki maroka samasya eduraiMdi. mana mEdhaki maroka varga rUpaM spuriMchiMdani anukuMdAM. I varga rUpaM tappO, oppO elA tElchaTaM? aMTE A rUpAnni upayOgiMchi pUrNa saMkhyalanniTini rAyagalamO lEmO elA tElchaTaM? pUrNa saMkhyalu anaMtaM kanuka idi saiddhAMtikaMgA nirNayiMcha valasinadE tappa prAyOgika paddhatulu paniki rAvu.

I praSna aparishkRtaMgA monna monnaTi varakU uMDi pOyiMdi. amerikAlO prins^Tan yUnivarSiTIlO unna maMjul bhArgava, DUk yUnivarsiTIlO unna jOnathan hen^kE tO kalasi paina cheppina jaTila samasyaki ati tElikaina samAdhAnaM uMdani rujuvu chEsEDu. bhArgava tanu sAdhiMchina parishkArAnni konni siddhAMtAla rUpaMlO, DiseMbaru 2005 lO, rAmAnujan janmasthalamaina kuMbhakONaMlO, SAstra viSvavidyAlayaMlO jarigina aMtarjAtIya samAvESaMlO vij~nula edaTa nirUpiMchi sabhikulani AScharya chakitulani chESATTa!

bhArgava bAlyaM nuMDI gaNitaMlO utsAhaM chUpeDutU vachchATTa. itanu 2001 lO prins^Tan lO pi. hech. Di. chEsE rOjulalO modalu peTTina pani punAdi anukuMTE kuMbhakONaMlO chadivina pariSOdhana patraM A punAdi mIda kaTTina mEDa. punAdula lOMchi I mEDa elA lEchiMdO oka nakha chitraMlA mI muMdu chitristAnu.

maLLA manaM charitralO koMcheM venakki veLLAli. jermanIlO 1801 lO mahA mEdhAvi kArl phrIDrIk gaus chEsina panini AdhAraMgA chEsukuni varga rUpAla mIda pariSOdhana modalu peTTEDu, mana bhArgava. gaus pani chEsina varga rUpAlu #ax2 + bxy + cy2# mAdiri uMTAyi. iTuvaMTi reMDu varga rUpAlani tIsukuni vATini saMdhiMchaTaM mIda konni saMdhi sUtrAlani (#composition laws)# pravachiMchEru gaus. saMdhiMchaTaM aMTE kalapaTaM lAMTi oka prakriyE kAni kalapaDaM kAdu. gaus pravachiMchina saMdhi sUtrAlE #algebriac number theory# anE oka kotta puMtaki mArgadarSi ayEyi.

prins^Tan lO vidyArdhi daSalOnE mana bhArgava iTuvaMTi saMdhi sUtrAlani marO padamUDiMTini kanukkunnADu. kanipeTTaTamE kAdu, gaNita SAstra rItyA I sUtrAlu elA udbhaviMchEyO kUDa rujuvutO sahA chUpeTTEDu. I pani phalitaMgA bhArgavaki paTTA ivvaTamE kAkuMDA 28 ELLa chiruta prAyAnikE AchArya padavi (#full professor)# ichchi gauraviMchiMdi, prins^Tan.

iMtakI bhArgava chEsiMdEmiTO cheppanE lEdu kadU? ‘E varga sUtraM upayOgiMchi pUrNa saMkhyalanniTini varNiMchagalaM?’ anna praSna bhArgavani modaTlO vEdhiMchaTaM modalu peTTiMdi. I rakaM varga rUpAlani viSvajanIna (lEdA sArvatrika) varga rUpAlu (#universal quadratic forms)# aMTAru.

gata SatAbdapu modaTi rOjullO rAmAnujan #a.x2 + b.y2 + c.z2 + d.t2# vaMTi rUpAlapai dRshTi kEdrIkariMchErani cheppukunnAmu kadA. Ayana I jAti rUpAlu 53 kanukkunnArani kUDA jAbitA vEsi chUpiMchEnu kadA. udAharaNaki #x2 + 2.y2 + 5.z2 + 10.t2# lO #x, y, z, t# la viluvalani mArchukuMTU pOtE dhana pUrNAMkAlanniTini sRshTiMchavachchu. udAharaNaki 14 kAvAlaMTE #x = 1, y = 2, z = 1, t = 0# ani pratikshEpistE saripOtuMdi. alAgE 32 kAvAlaMTE #x = 0, y = 2, z = 2, t = 1# pratikshEpiMchAli.

‘iMkA ilAMTi sUtrAlu enni unnAyi?’ anna praSnaki samAdhAnaM cheppAlanukuMTE, manaki suLuvaina parIksha okaTi kAvAli.

sA. Sa. 1993 lO prins^Tan yUnivarsiTI lO pani chEsE jAn kAnvE anE AchAryuDu tana daggara pani chEsE vidyArdhi viliyaM shIMbergar tO kalasi aTuvaMTi viSvajanIna varga rUpAnni oka dAnini pratipAdiMchEDu. I rUpaM oka mAtruka (#matrix)# rUpaMlO rAsEru vAru. I rUpAnni upayOgiMchi 1, 2, 3, 5, 6, 7, 10, 14, 15 anE tommidi saMkhyalani utpatti cheyyagaligitE, migilina pUrNa saMkhyalanniTini kUDa utpatti cheyyagalaM anE sUtrAnni vAriruvuru ‘rujuvu’ chEsEru. idE 15-siddhAMtaM anE pErutO chelAmaNI kAvaTaM modaleTTiMdi.

lOgaDa manaM chUsina lagrAMj sUtrAlu, rAmAnujan sUtrAlU kUDa I 15-siddhAMtAniki lObaDE uMTAyi kanuka, sUtra bhaMgAlEmI kAlEdani aMdarU oka sAri tElikagA Upiri pIlchukunnAru. ayinA sarE kAnvE prabhRtulu vAri siddhAMtAnnI, dAnini rujuvu chEsE saMklishTamaina paddhatinI ekkaDA prachuriMcha lEdu. ilA prachuriMchakuMDA uMDaTAniki sAdhAraNaMgA reMDu kAraNAlu uMTAyi. okaTi, siddhAMtaMlO EmainA losugulu uMTE paruvu pOtuMdanE bhayaM. reMDu, siddhAMtaM anuvartiMchE vyAptini peMchi appuDu prachuriddAmulE anE saduddESaM. aMdukani vAri sUtraM anuvartiMchE paridhini peMchaTAniki pariSOdhana modalu peTTEru. I pariSOdhanalO vAru maroka varga rUpAnni kanukkunnAru. I rUpamE 3.#x2 + xy + 5.y2 + 6.z2 + t2. #“I rUpaM upayOgiMchi 1 nuMDi 290 varaku unna anni saMkhyalani utpatti cheyya galigitE I sUtrAnni viSvajanIya varga sUtraMgA parigaNiMchavachchu” ani oka UhAgAnaM chesEru. kAni rujuvu cheyya lEdu (lEdA, rujuvu cheyya lEkapOyi uMDochchu kUDA).

I paristhitilO bhArgavaki kAnvE I 15-siddhAMtAnni parichayaM chEsEru. “kAnvE cheppina kathanaM vinna tarvAta nAku nOTa mATa rAlEdu. gaNitaMlO iTuvaMTi phalitaM uMdani telisE sariki AScharyaM veyyaTaM oka ettayitE, I phalitaM rujuvu lEkuMDA kEvalaM UhAgAnaMlA uMDipOyiMdani teliyaTaM maroka ettu” ani bhArgava vyAkhyAniMchi, “venu veMTanE nEnu chEstUnna panulannI ApEsi I UhAgAnAniki rujuvu vetakaTaM modalu peTTEnu,” annADuTa.

bhArgava 15-siddhAMtAniki oka kotta paMthAlO rujuvuni nirmichaTaM modalupeTTEDu. I kotta dAri veMbaDi veLitE rujuvu cheyyaTaM tElikavaTamE kAkuMDA, chAlA takkuva jAgAlO rujuvu cheyyaTAniki vIlayiMdiTa. I rujuvu prakAraM mottaM 204 (mAtruka rUpaMlO nirvachiMchabaDDa) viSvajanIna varga rUpAlu unnAyiTa.

I rujuvu gaNita prapaMchAnni adaragoTTiMdi. eMdukaMTE sA. Sa. 1948 lO mArgareT villarDiMg idE praSnani edurkoni, aharniSalu kashTapaDi 178 viSvajanIna varga sUtrAlu kanukkunnAru. bhArgava chEsina pani nEpadhyaMlO Ame kanibeTTina 178 sUtrAlalO okE sUtraM porapATuna reMDu sArlu dorliMdaninnI, 9 sUtrAlu pUrtigA tappaninnI telisiMdi. mukhyamaina vishayaM EmiTaMTE bhArgava ichchina rujuvulu chinnavi gAnU, ardhaM ayE rItilOnU unnAyiTa. purushulalO puNya purushulu unnaTlE rujuvulalO aMdamaina rujuvulu uMTAyi. sUTigA, saMkshiptaMgA unna rujuvulU, siddhAMtAlU, sUtrAlU aMdamaina vATigA lekka.

I 15-siddhAMtAniki rujuvu kanukkunna tarvAta bhArgava 33-siddhAMtaM ani marO siddhAMtaM kanukkunnAru. I sUtraM 1, 3, 5, 7, 11, 15, 33 saMkhyala eDala panichEstE migilina anni bEsi saMkhyala eDala kUDA panichEstuMdani I 33-siddhAMtaM yokka sArAMSaM. I siddhAMtAnni bhArgava rujuvu chEsina vainaM chUsi “idi chAla aMdamaina rujuvu” ani kAnvE abhivarNiMchEru Ta.

idE dhOraNilO bhArgava pradhAna saMkhyalu (#prime numbers)# anniMTini utpatti chEyagala varga rUpAnni okadAnini nirmiMchEru.

I pAvaMchAlannI dATukuni kAnvE 290 guriMchi pratipAdiMchina UhAgAnAniki kUDA bhArgava, hen^ke kalasi rujuvu chUpiMchEru. idoka pedda mailu rAyi. kanuka ippuDu manaki varga sUtrAla yokka svarUpa svabhAvAlu paripUrNaMgA avagAhana ayinaTlE – ani anukuMTunnAM, prastutAniki. vIru cheppEdi EmiTaMTE – E varga rUpamainA sarE pUrNa saMkhyalanniTini utpatti cheyyagaladO lEdO nirNayiMchAlaMTE muMdu A rUpaM 290 tOpATu 290 ki lOpugA unna oka 29 pUrNAMkAla samitini utpatti cheyyagaladO lEdO chUDAliTa. I samiti (#set)# lO unna 29 pUrNAMkAlanI utpatti cheyyagalaM ani telisina mIdaTa alA utpatti cheyyagaligE varga rUpAlu 6,436 unnAyani rujuvu chEsEru!

idE vishayAnni bhArgava kuMbhakONaMlOni samAvESaMlO chebitE Ayanaki rAmAnujan smAraka chihnamaina ‘SAstra’ patakAnni ichchi gauraviMchEruTa.

I rakaM lekkala prayOjanaM EmiTi ani chAla maMdi pedavi virustAru. aMdame AnaMdaM annAru. dInnE iMglIshulO #A thing of beauty is a joy for ever# aMTAru. kanuka I rakaM rujuvula kOsaM vetakaTaM oka rakamaina sauMdaryOpAsana. panikimAlina I rakaM upAsanalu evvariki kAvAli ani tOsi puchchakaMDi. I kaMpyUTar yugaMlO #cryptography# ki prAmukhyata perugutOMdi. I raMgaMlO rAmAnujan vaMTi vAru chEsina pariSOdhanalu upayOgapaDutunnAyi. idE vidhaMgA viSva rahasyAlani ChEdiMchaTAniki vADE gaNitaMlO kUDA rAmAnujan prabhAvaM kanabaDaTaM modalaiMdi.

AdhAraM: #Science News,# mArchi 11, 2006 saMchikalO #Ivars Peterson# rAsina #All Square# anE vyAsaM dIniki AdhAraM (#Translated with permission from Ivars Peterson and Science Service. Copyright by Science Service).

    
   
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(11# abhiprAyAlu) mI abhiprAyaM teliyachEyaMDi #»

  1. rajay # abhiprAyaM:

    #September 1, 2006 9:41 pm

    Good article on maths sir.
    Keep it up.
    cheers.
    rajay
    sgp

  2. Rama S Yeleswarapu # abhiprAyaM:

    #September 6, 2006 12:45 pm

    #ayyA,
    bhArgava gAri pariSOdhani chAlA aMdaMgA adhbutaMgA vrASAru vEmUri gAru. I vyAsaM chadivina taruvAta gaNitaM mIda nAku marala utsukata kaligiMdi.
    dayachEsi iTuvaMTi vyAsAlu marinni bhavishyattu lO prachuriMcha prArdhana.

    bhavadIyuDu,
    rAma subrahmaNyaM yElESvarapu.

  3. #Kiran Kumar Chava # abhiprAyaM:

    #September 10, 2006 9:24 pm

    #maMchi vyAsaM

    nAkkUDA maraLA palakA balapaM paTTukOni I rujuvulU, sUtrAlU arthaM chEsukOvAlanipistuMdi

    telugulO ilA Adhunika vyAsAlu vastE mast utsAhaMgA uMTuMdanukuMTa

  4. koDavaLLa hanumaMtarAvu abhiprAyaM:

    #September 16, 2006 1:01 pm

    “#panikimAlina” lekkala prAdhAnyata

    telugulO SAstrIya vishayAlni iMta tEliggA teliyaparichina vEmuri vyAsaM chAlA bAvuMdi. “peddavALLu pustakAlu rAstAru, yuvakulu siddhAMtAlani rujuvu chEstAru,” anna sUtrAnni bhArgava marOsAri rujuvu chESAru.

    ilAMTi lekkala sogasulni chUDAlannA,
    “#Not all the water in the rough rude sea
    Can wash the balm off from an anointed king;”
    #kannA
    “#After life’s fitful fever he sleeps well; ”
    #eMduku goppa kavitvamO telusukOvAlannA, tappaka chadavAlsina pustakaM, rAmAnujan ni kanugonna #Hardy# rAsina “#A Mathematician’s Apology.”# achchupratilO unna #C. P. Snow# rAsina muMdumATa gUDA chadavadaggadi.

    koDavaLLa hanumaMtarAvu

  5. rAkESvara rAvu abhiprAyaM:

    #April 14, 2008 3:10 am

    #vEmUri vAriki kRtaj~natalu
    rAkESvara

  6. brahmESvara rAvu mEkA abhiprAyaM:

    #January 30, 2014 1:38 am

    #iTu vaMTi vyAsAlu aMdarikI chErAli. vEmUri vAriki, I mATa ku kRtaj~natalu.

  7. #Rao Vemuri # abhiprAyaM:

    #August 13, 2014 9:45 am

    #I vyAsaMlO uTaMkiMchina maMju bhArgavaki nObel^ bahumAnaMtO tulatUgE “phIl^Ds^ meDal^” vachchiMdanna vArta chadavagAnE aMdaritOTI paMchukOvAlanipiMchiMdi. – vEmUri

  8. vAsudEva rAvu erikalapUDi abhiprAyaM:

    #August 15, 2014 10:00 am

    #SrI vEmUri: enimidi saMvatsarAla kritaM yeMtO kashTamaina vishayaM mIda mIru rAsina I chakkani vyAsaM iMtaku muMdu chadivina gurtu rAvaTaM lEdu. maMjul bhArgavaki I gauravaM labhiMchaTaMlo mI AnaMdaMlo nEnU pAlupaMchukuMTunnAnu. AMgla dinapatrika hiMdUlo I roju vachchina iMTarvyU lO bhArgava tanaku priyamaina ‘mUDu dESAla saMskRti’ni vivariMchaTaM aMdarU chadavavalasinadi.

    namaskArAlatO
    vAsudEva rAvu

  9. vishNubhoTla lakshmanna abhiprAyaM:

    #August 15, 2014 2:30 pm

    Here is the link to the Hindu article on Majul Bhargava:

    Major mathematics awards to two Indian origin scientists

    #vishNubhoTla lakshmanna

  10. vAsudEva rAvu erikalapUDi abhiprAyaM:

    #August 17, 2014 4:33 am

    #nA muMdaTi abhiprAyaMlO uTaMkiMchina iMTarvyU ‘ikkaDa’ chadava vachchu.

    vAsudEva rAvu

  11. #Rao Vemuri # abhiprAyaM:

    #August 20, 2014 9:54 am

    #vAsudEvarAvu gArU,
    mI I-TapA chirunAmA nAku (Ar^vEmUri eT^ jimail^) paMpitE mIku nEnu I madhya prachuriMchina “nityajIvitaMlO rasAyanaSAstraM” anE I-pustakaM paMpagalanu. – vEmUri

mI abhiprAyaM teliyachEyaMDi

#  

  

#ImATa pAThakulaku sUchanalu chadivAnu. vATiki kaTTubaDi uMTAnani hAmI istunnAnu.