daayofainteeya sameekaran

poornaank bhujaaon vaale sabhi samakon tribhuj praapt karna ek prakaar se daayofainteeya sameekaran ke hal karne ke tulya hi hai.

daayofaintas naamak yoonaani ganitjnya ne, jo sambhavat: isa ke pashchaat teesari shataabdi mein raha, bahut se bahupadeeya anirdhaarya sameekaranon (Undetermined Equations) ka adhyayan kiya tatha poornaankon mein unke halon ko gyaat kiya. kintu aadhunik tathyon ke prakaash mein ab ismein koi sandesh naheen rah gaya hai ki daayofaintas se bahut pehle hi bhaarateeya ganitjnyaon ne is kshetr mein atyant mahatvapoorn kaarya kar liya tha.

daayofainteeya samasyaaon mein sameekaranon ki sankhya charon ki sankhya se kam hoti hai (arthaat sameekaran anirdhaarit hote hain). in sameekaranon mein charon ke gunaank poornaank hote hain aur charon ka aisa poornaank maan praapt karna hota hai jo in sameekaranon ko santusht karein. isliye un sabhi sameekaranon ka naam daayofainteeya sameekaran (Diophantine Equations) pad gaya jinmein charon ke gunaank poornaank hote hain.

daayofainteeya samasyaaon ke ganiteeya adhyayan ko aajkal daayofainteeya vishleshan (Diophantine analysis) kehte hain. itihaas mein harek daayofainteeya sameekaran ek bujhauval (puzzle) ki tarah prayog ki jaati rahi hain kintu daayofainteeya sameekaranon ka saamaanya siddhaant (jo dvighaat roopon ke pare bhi jaanya) beesaveen shati ki ek badi ganiteeya upalabdhi maani jaati hai.

anukram

daayofainteeya sameekaran ke udaaharan

neeche diye gaye daayofainteeya sameekaranon mein x, y, tatha z agyaat raashiyaaain hain; anya akshar niyat raashiyaaain hain.
yeh ek raikhik daayofainteeya sameekaran hai.
n = 2 ke liye (x,y,z) ke anant hal sambhav hain jo paaithaagoreeya trik (Pythagorean triple) honge. n ke 3 se bade maanon ke liye, farma ke antim prameya (Fermat's Last Theorem) ke anusaar (x, y, z) ka koi bhi dhanaatmak poornaank hal sambhav naheen hai
yeh pell ka sameekaran hai jo angrej ganitjnya John pell (John Pell) ke naam par pada hai. ise brahmagupt ne saataveen shati mein iska adhyayan kiya tha. farma (Fermat) ne satrahaveen shati mein iska adhyayan kiya.
yeh irdaus-straus kaanjekchar (Erdőas–Straus conjecture) kahalaata hai. iske anusaar, 2 se badi dhan poornaankon ke liye (n ≥ 2), ek hal sambhav hai jismein x, y, tatha z sabhi dhan poornaank honge. yadyapi is sameekaran ko bahupad ke roop mein naheen vyakt kiya jaata, kintu yeh udaaharan bahupad sameekaran 4xyz = yzn + xzn + xyn = n(yz + xz + xy) ke tulya hi hai.

daayofainteeya vishleshan

daayofainteeya vishleshan mein praaya: nimnalikhit prashnon par vichaar kiya jaata hai-

  • kya koi hal sambhav hai?
  • kisi di hui sankhya (jiske liye aasaani se dekhkar hi pata chal jaata hai ki yeh daayofainteeya sameekaran ka hal hai) ke pare koi hal sambhav hain?
  • halon ki sankhya seemit hai ya anant ?
  • kya saiddhaantik roop se sabhi hal praapt karna sambhav hai?
  • kya koi vyakti sabhi halon ki ganana kar sakta hai? kya yeh vyaavahaarik hai?

Bhaarat ke praacheen ganitjnyaon dvaara krut daayofainteeya vishleshan

brahmagupt

800 isaapoorv se 500 isaapoorv rachit shulbasootron mein daayofainteeya sameekaranon ka poornaank hal milta hai. lagbhag 800 isaapoorv mein baudhaayan ne yugapat (simultaneous) daayofainteeya sameekaranon ka hal do dhanaatmak poornaakon ka set ke roop mein praapt kiya tha. unhone chaar agyaat raashiyon se yukt yugapat daayofainteeya sameekaranon ka hal nikaalne ki bhi koshish ki thi. 600 isaapoorv mein aapastamb ne 5 agyaat raashiyon vaale daayofainteeya sameekaranon ka hal karne ki koshish ki thi.[1]

kuttak

ati praacheen kaal se hi bhaarateeya ganitjnya prakaar ke sameekaranon ka poornaank hal khojane ki disha mein kaaryarat rahe hain. yadyapi aajkal inhein 'daayofainteeya sameekaran' kaha jaata hai kintu daayofaintas se bahut pehle shulbasootron mein aisi ganiteeya samasyaaon ki vishd charcha hai. shulbasootron ka kaal 800 isaapoorv ya usase bhi pehle hai.

aaryabhat ne is prakaar ki samasyaaon ke hal ko 'kuttak' naam diya hai. bhaaskar ne aaryabhateeya ki teeka mein nimnalikhit kuttak diya hai-

vah sankhya nikaalo jismein 8 se bhaag dene par 5 shesh rahata hai, 9 se bhaag dene par 5 shesh rahata hai tatha 7 se bhaag dene par 1 shesh rahata hai.

arthaat poornaank nikaalo jahaaain . ise hal karne par ka maan aata hai.

'kuttak' ka shaabdik arth 'kootane wala' (palvaraaijar) hai. kuttak vidhi vaastav mein ek punaraavarti (recursive) vidhi hai. vartamaan samay mein bhaaskar dvaara 621 E mein vyaakhya sahit di gayi is vidhi ko 'aaryabhat ki kalanavidhi' (Aryabhata algorithm) kehte hain jo pratham ghaat vaale daayofainteeya sameekaranon ke hal ki maanak vidhi ban gayi hai.

isi prakaar, brahmagupt ne nimnalikhit anirdhaarya sameekaran ko hal kiya hai:

brahmagupt ke 'khandakhaadyak' naamak granth mein 'kuttakaadhyaaya' naamak ek poora adhyaaya hai.

17veen evam 18veen shtaabdi

farma (Pierre de Fermat) ne is disha mein kaafi kaam kiya.

hillbart ka 10vaaain prashn

aadhunik anusandhaan

sandarbh

baahari kadiyaaain