
vibhajana yokka crank

sankhya siddhaantamlo, oka puurnaamkam yokka vibhajana yokka crank anedi vibhajanatho anubandhinchabadina oka nirdishta puurnaamkam. eee padaanni 1944loe cambridge vishwavidyaalayam yokka ganita sangham prachurinchina journal eureka loe prachurinchina oka paperlo freeman dyson nirvachanam lekunda modatisaarigaa pravesapettaaru. dyson appudu eee inca nirvachinchabadani parimaanam kaligi undaalsina lakshanaala jaabitaanu ichindi. 1988loe, georg ii. andrus mariyu fraank garvan dyson dwara oohinchina lakshanaalanu santruptipariche crank kosam oka nirvachananni kanugonnaru.
dyson yokka crank.
n" anedi oka pratikuulam kaani puurnaamkam mariyu "p" ("n") anedi "n" ("p" (0) yokka vibhajanala sankhyanu 1 gaa nirvachinchabadutundi. 1918loe prachurinchabadina oka patrikalo srinivasa ramanujan vibhajana function "p" ("n") kosam eee krindi saaruupyatalanu paerkonnaadu mariyu niroopinchaadu, dheenini ramanujan saaruupyaalu anie pilustaaru.
eee saaruupyatalu 5 "n" + 4 (varusaga, 7 "n" + 5 mariyu 11 "n" + 6 roopaala) sankhyala vibhajanalanu samaana parimaanam gala 5 (varusaga, 7 mariyu 11) upavargaalugaa vibhajinchavacchani suuchistaayi. eee sammelanaala yokka appati telisina rujuvulu vidhulanu utpatti chese aalochanalapai aadharapadi undevi mariyu avi vibhajanalanu samaana parimaanam gala upavargaalugaa vibhajinchadaaniki oka paddhatini perkonaledu.
thana eureka paper dyson loe vibhajana yokka rank yokka bhavananu pratipaadinchaadu. vibhajana yokka rank anedi vibhajanaloni atipedda bhaagam nundi vibhajanaloni bhagala sankhyanu teesiveyadam dwara pondina puurnaamkam. udaaharanaku, 9 yokka vibhajana λ = {4,2,1,1} yokka rank 4−5 = −1. "n" ("m", "q", "n") dwara suuchimchabadutumdi, "n" yokka vibhajanala sankhya, dheeni ranklu "m" maadyulo "q kuu samanamga untaayi, dyson" n "(" m ", 5", n "+ 4) mariyu" n "(" m ", 7", n "+ 5) vividha viluvalaku" n "mariyu" m ". anubhavika saakshyaala aadhaaramgaa dyson "rank consecturers" anie piluvabade krindi consecturesnu roopondinchindi.
anni pratikuula kaani puurnaamkaalu "n" kosam manaku unnaih
eee oohagaanaalu nijamani bhaavinchi, avi 5 "n" + 4 roopamlo unna anni vibhajanalanu samaana parimaanamlooni aidu taragatulugaa vibhajinche marganni andinchayih oka taragatilo unchandi, aaa vibhajanalannintini, vaati shrenulu okadaanikokati maadyulo 5kii samanamga untaayi. 7 "n" + 6 roopamlo unna puurnaamkaala vibhajanalanu edu samanamga aneka taragatulugaa vibhajinchadaaniki adhe aalochananu vartimpajeyavachu. conny eee krindi pattika chuupinatlugaa, 11 "n" + 6 roopam yokka puurnaamkaala vibhajanalanu adhe parimaanam yokka 11 taragatulugaa vibhajinchadamlo aalochana viphalamaindi.
anduvalla siddhaantaanni samanvayamgaa niroopinchadaaniki ranknu upayoginchalemu. ayithe, dyson ilaa vraasaadu,
"neenu nizaniki"
"eee anchanaalu sakshyam dwara haamii ivvabadataya, neenu daanini nirnayinchadaaniki paatakulaku vadilivestaanu. bhavishyat yokka tudi teerpu emainappatiki, "crank" anedi kanugonabadataaniki mundu paeru pettabadina ganita vidhulalo pratyekamainadani neenu nammutunnaanu. valkan graham yokka avamaanakaramaina vidhi nundi idhi samrakshinchabadavachhu. "
crank yokka nirvachanam.
1988loe georg ii. loe prachurinchabadina oka patrikalo. andrus mariyu eff. zii. garvan vibhajana yokka kraanknu eee krindi vidhamgaa nirvachinchaaruh
4, 5, 6 puurnaamkaala vibhajanala cranklu eee krindi pattikalalo lekkinchabadataayi.
sankethaalu.
n" (−1) = − "m" (0,1) = "m" (1,1) = 1 krindi utpatti function dwara ivvabadina vidhamgaa "n" ≤0 mariyu anni puurnaamkaalu "m" kosam, "m" kuu samaanamaina crank thoo "n" yokka vibhajanala sankhyanu "m" ("m", "n") dwara suuchistaaru. m" maadyulo "q" kuu samaanamaina crank thoo "n" yokka vibhajanala sankhyanu "m" ("m", "q", "n") dwara suuchistaaru.
m" ("m", "n") kosam utpatti chese function krinda ivvabadindi
praathamika phalitam.
andrewlu mariyu garvan eee krindi phalitaanni niroopinchaaru, idhi paina nirvachinchina vidhamgaa crank dyson icchina paristhitulaku anugunamgaa undani choopistundi.
rank mariyu crank yokka bhaavanalanu konni puurnaamkaala vibhajanalanu samaana parimaanam gala upavargaalugaa vargeekarinchadaaniki upayoginchavacchu. ayithe eee rendu bhavanalu vibhajanala yokka ververu upavargaalanu utpatti chestaayi. idhi eee krindi rendu pattikallo vivarinchabadindi.
ramanujan mariyu cranks.
broose sii iteevala chesina pani. berned mariyu athani saha rachayitalu ramanujanku crank gurinchi telusani velladinchaaru, ayithe andrus mariyu garvan nirvachinchina roopamlo kaadu. ramanujan yokka kolpoyina notebook yokka kramabaddhamaina adhyayanamlo, berned mariyu athani saha rachayitalu ramanujanku crank generating function yokka vichchaedala gurinchi telusani gananeeyamaina aadharalu icchaaru.

