
pathyakram-moolya punaravritti

gananashilata siddhant mein, pathyakram-moolya punaravritti punaravritti dwara sankhya-saiddhaantik karyon koo paribhaashit karane key liye ack takneek haye. ack falan "f" kii paribhasha mein pathyakram-maan punaravritti dwara, "f" ("n") kaa moolya anukram sootra _ 1 seey ganana kii jaati haye. 
iss tathya kaa kii aisee paribhashaon koo punaravritti key ack saral roop kaa upayog karke paribhashaon mein parivartit kiya jaa sakata haye, aksar yeh saabit karane key liye upayog kiya jaataa haye kii pathyakram-moolya punaravritti dwara paribhaashit kaary adim punaraavarti hain. pathyakram-maan punaravritti key vipareet, adim punaravritti mein ack falan key moolya kii ganana key liye kewal pichhale moolya kii aavashyakta hoti haye; udaaharan key liye, ack 1-arrie adim punaraavarti falan "g" key liye "g" ("n" + 1) kaa moolya kewal "g" ("n") aur "n" seey ganana kii jaati haye.
paribhasha aur udaaharan.
factorial function "aine"! niyamon dwara punaraavartit roop seey paribhaashit kiya jaataa haye
yeh punaravritti ack adim punaravritti haye kyonki yeh agale moolya ("n" + 1) kii ganana kartaa haye! n" key moolya aur pichhale moolya "n" key aadhar parr falan kaa! kaary kaa. doosari or, falan fibe ("aine"), joe "aine" th fibonacci sankhya detaa haye, koo punaravritti samikaranon key saath paribhaashit kiya jaataa haye.
fibe ("aine" + 2) kii ganana karane key liye, fibe function key antim "dou" maanon kii aavashyakta hoti haye. anta mein, punaravritti samikaranon key saath paribhaashit falan "g" parr vichaar karein.
inn samikaranon kaa upayog karke "g" ("n" + 1) kii ganana karane key liye, "g" key sabhi pichhale moolyon kii ganana kii jaani chaahiye; "g" kii ganana key liye samanya roop seey pichhale moolyon kii koee nishchit seemit sankhya paryaapt naheen haye. falan fiber aur "jee" pathyakram-maan punaravritti dwara paribhaashit karyon key udaaharan hain. 
samanya taur parr, ack falan "f" koo pathyakram-maan punaravritti dwara paribhaashit kiya jaataa haye yadi ack nishchit adim punaraavarti falan "h" aisaa haye kii sabhi "n" key liye, 
jahan sootra _ 10 ack godel sankhya haye joe ingit anukram koo kootbaddh karati haye.
vishesh roop seey
punaravritti kaa praarambhik moolya pradaan kartaa haye. falan "aich" spasht praarambhik moolya pradaan karane key liye apane pahale tark kaa pareekshan curr sakata haye, udaaharan key liye fiber key liye ack dwara paribhaashit kaary kaa upayog curr sakata haye
jahan "s" ["i"] ack kootbaddh anukram "s" seey tatva "i" key nishkarshan koo darshaata haye; yeh aasaani seey ack adim punaraavarti kaary key roop mein dekhaa jaataa haye (yeh maanate hue kii ack upayukt godel sankhya kaa upayog kiya jaataa haye).
adim punaravritti key liye samatulyata.
pathyakram-moolya punaravritti dwara ack paribhasha koo ack adim punaravritti mein badalane key liye, ack sahayak (sahayak) kaary kaa upayog kiya jaataa haye. maan leejiye kii koee chaahata haye
adim punaravritti kaa upayog karke paribhaashit karane key liye, pahale sahayak pathyakram-moolya kaary koo paribhaashit karein jise santusht karna chaahiye
jahan daahine haath kii or anukramon key liye ack godel sankhya maanaa jaataa haye.
iss prakaar sootra _ 15 key pahale maanon koo kootbaddh kartaa haye. falan sootra _ 16 koo adim punaravritti dwara paribhaashit kiya jaa sakata haye kyonki sootra _ 17 sootra _ 15 mein nae tatva sootra _ 19 koo jodkar praapt kiya jaataa haih
jahan ganana kii jaati haye, jaba bhee lambai kaa ack anukram kootbaddh kiya jaataa haye, lambai kaa ack nayaa anukram aisaa aur sabhi key liye. yeh ack upayukt godel sankhya kii dhaarana key tahat ack adim punaraavarti kaary haye; "aich" koo shuru mein adim punaraavarti maanaa jaataa haye. iss prakaar punaravritti sambandh koo adim punaravritti key roop mein likhaa jaa sakata haih
jahan "g" swayam hii adim punaraavarti haye, aise dou karyon kii sanrachana honey key kaaranah
sootra _ 16 diye jaane parr, mool falan koo sootra _ 25 dwara paribhaashit kiya jaa sakata haye, joe darshaata haye kii yeh ack adim punaraavarti falan bhee haye.
adim punaraavarti karyon key liye anuprayog.
adim punaraavarti karyon key sandarbh mein, praakritik sankhyaon key seemit anukramon koo ekal praakritik sankhyaon key roop mein prastut karane kaa ack saadhan honah suvidhajanak haye. aisee hii ack vidhi, godel kaa encoding, sakaratmak purnank sootra _ 26 key anukram kaa pratinidhitva karati haye.
jahan "p" "i" "i" th prime kaa pratinidhitva kartaa haye. yeh dikhaaya jaa sakata haye kii, iss pratinidhitva key saath, anukramon parr samanya sanchaalan sabhi adim punaraavarti hain. inn karyon mein shamil hain
anukramon key iss pratinidhitva kaa upayog karate hue, yeh dekhaa jaa sakata haye kii yadi "aich" ("aim") adim punaraavarti haye tou falan
yeh bhee adim punaraavarti haye.
jaba anukram sootra _ 26 koo shoonya koo shamil karane kii anumati dee jaati haye, tou ise isake bajaay key roop mein darshaaya jaataa haye
joe anukram sootra _ 31 aur sootra _ 32 key liye code koo alag karna sambhav banaata haye.
seemaaen.
pratyek punaraavarti paribhasha koo ack adim punaraavarti paribhasha mein naheen badala jaa sakata haye. ack gyaat udaaharan ackerman kaa kaary haye, joe "aeye" ("aim", "aine") key roop mein haye aur yeh nishchit roop seey adim punaraavarti naheen haye.
vaastav mein, pratyek nayaa maan "a" ("m" + 1 ", n") pahale seey paribhaashit moolyon "a" ("i", "j") key anukram parr nirbhar kartaa haye, lekin "i"-s aur "j"-s jinake liye moolyon koo iss anukram mein shamil kiya jaanaa chaahiye, way swayam kaary key pahale seey sanganit moolyon parr nirbhar karate hain; arthaat ("i", "j") = ("m", "a" ("m" + 1 ", n")). iss prakaar koee bhee moolyon key pahale seey sanganit anukram koo uparokt sujhaae gaye tareeke seey adim punaraavarti tareeke seey kootbaddh naheen curr sakata haye (yaa bilkul bhee, kyonki yeh pataa chalata haye kii yeh kaary adim punaraavarti naheen haye).

