sankhya siddhaant

yeh lekh sankhya paddhati (number system) ke baare mein naheen hai.


chitr:Lehmer sieve.jpg
lemar chalni (A Lehmer sieve), jo 'aadim computer' kahi ja sakti hai. kisi samay isi ka upayog karke abhaajya sankhyaaeain praapt ki jaateen theen tatha saral daayofainteeya sameekaran hal kiye jaate the.

sankhya siddhaant (Number theory) saamaanyat: sabhi prakaar ki sankhyaaon ke gunadharm ka adhyayan karta hai kintu visheshat: yeh praakrutik sankhyaaon 1, 2, 3....ke gunadharmon ka adhyayan karta hai. poornata ke vichaar se in sankhyaaon mein ham rin sankhyaaon tatha shoonya ko bhi sammilit kar lete hain. jab tak nishchit roop se na kaha jaae, tab tak sankhya se koi praakrutik sankhya, dhan, ya rin poorn sankhya ya shoonya samajhna chaahiye. sankhyaasiddhaant ko gaaus (Gauss) ganit ki raani kehta tha. sankhya siddhaant, shuddh ganit ki shaakha hai.

'sankhya siddhaant' ke liye "ankaganit" ya "uchch ankaganit" shabdon ka bhi prayog kiya jata hai. ye shabd apekshaakrut puraane hain aur ab bahut kam prayog kiye jaate hain.

anukram

parichay

is ganit ki raani ke anupam gunon mein se ek gun, jiske kaaran chhote-bade sabhi prakaar ke ganitjnya iski or aakarshit hue hain, yeh hai ki sankhya siddhaant ke anek prashn saadhaaran vidyaalayon ke vidyaarthiyon ki samajh mein to aa jaate hain, parantu hal karne mein ve itne saral naheen hain. udaahanasvaroop, goldabaik ke anumaan (Goldbach's Conjecture) ko lein, jiske anusaar 2 se badi pratyek sam sankhya, do abhaajyon ke yogafal ke roop mein niroopit ki ja sakti hai. is anumaan ka satyaapan to bahut adhik ho gaya hai, parantu abhi tak iska siddh karne mein, ya isko asatya karne mein kisi ganitjnya ko safalta naheen mili hai. iske vipreet ek hi udaaharan isko asatya thaharaane ke liye paryaapt hoga, jab ki ise paksh mein laakhon udaaharan iski satyata ko siddh thaharaane ke liye paryaanpt naheen ho sakte. vinogredov (Vinogradov) ki vidhi se ham is anumaan ke nikat pahunchate hain. yeh siddh kiya ja chuka hai ki sab badi visham sankhyaaeain teen abhaajyon ke yogafal hain.

yadi koi sankhya yadruchhaya (at random) di gayi hai, to saamaanya: yeh kehna sambhav naheen hai ki vah sankhya abhaajya hai athva naheen. yadi di hui sankhya badi sankhya hai, to iski jaaainch mein bahut shram karna padega. is shram ko kam karne ki kai vidhiyaaain nikaali gayi hain, parantu samasya jyon ki tyon bani hui hai.

shaakhaaeain

aarambhik sankhya siddhaant

aarambhik sankhya siddhaant mein ganit ki doosari shaakhaaon ka sahaara liye bina hi poornaankon ke gunon ka adhyayan kiya jaata hai. iske antargat vibhaajyata, mahattam samaapavartak nikaalne ke liye prayukt yooklid ka algoridm, sankhyaaon ke abhaajya gunakhand nikaalna, poorn sankhyaaon (parafekt nambars) tatha samasheshata (congruences) aadi ka adhyayan kiya jaata hai.

vaishleshik sankhya siddhaant

vaishleshik sankhya siddhaant mein poornaakon ka adhyayan karne ke liye kailakulas tatha samishr vishleshan (kampeks enaalisis) kaki sahaayata li jaati hai. bhaajya sankhya prameya (prime number thiaram) iska ek udaaharan hai.

beejeeya sankhya siddhaant

beejeeya sankhya siddhaant (algebraic numer theory) mein adhyayan ki jaane waali sankhyaaon ka aur adhik saamaanyeekaran kiya gaya hai aur keval poornaankon ke gunon ka adhyayan karne ke sthaan par 'beejeeya sankhyaaon' ke gunon ka adhyayan kiya jaata hai. koi bhi sankhya jo kisi poornaank gunaakon vaale ekachareeya bahupadeeya sameekaran ka mool ho use 'beejeeya sankhya' kehte hain.

jyaamiteeya sankhya siddhaant

ismein sabhi prakaar ki jyaamitiyon ka upayog hota hai. ferma ka antim prameya isi vidhi se siddh kiya gaya tha.

abhiklani sankhya siddhaant

abhiklani sankhya siddhaant (computational number system) ke antargat sankhya siddhaant ke liye upayogi elgoridmon ka adhyayan kiya jaata hai. udaaharan ke liye, abhaajyata siddh karne vaale daksh kalan vidhiyon ka vikaas tatha sankhyaaon ke abhaajya gunakhanad nikaalne ki vidhiyaaain aadi.

anuprayog

1974 mein donaald nuth (Donald knuth) ne kaha tha ki kampyootaron se teevr gati se gananaaeain karaane ki koshish mein praarambhik sankhya siddhaant ke lagbhag sabhi prameyon ki aavashyakta pad jaati hai. computer vigyaan ke paathyakram mein vivikt ganit ke antargat sankhya siddhaant bhi padhaaya jaata hai. in sabke alaava beej-lekhan (kriptograafi) aur satat aankik vishleshan mein bhi sankhya siddhaant ka upayog kiya jaata hai.

inhein bhi dekhein

baahari kadiyaaain