Pythagorean triples kya hain? (Baudhayana triples)

Poori sankhya wali bhujaon ke right triangles

Zyaadatar right triangles ki kam se kam ek bhuja poori sankhya nahi hoti: 1 aur 1 bhujaayein √2 hypotenuse deti hain. Par kuch ki teeno bhujaayein poori sankhyaayein hoti hain. Apne Shulbasutra (verse 1.13) mein Baudhayana ne chhah likhe: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (15, 36, 39) aur (12, 35, 37). Jin teen ginti ki sankhyaon mein a² + b² = c² ho, unhe Baudhayana triples kehte hain, aur Baudhayana-Pythagoras triples ya Pythagorean triples bhi. Ek jaancho: 5² + 12² = 25 + 144 = 169 = 13².

Triple ko scale karna

Kya (30, 40, 50) triple hai? Haan: 900 + 1600 = 2500. (300, 400, 500) bhi. Aam taur par har ginti ki sankhya k ke liye (3k, 4k, 5k) chalta hai, kyunki 9k² + 16k² = 25k². Yahi tark har triple par lagta hai: agar a² + b² = c², to (ka)² + (kb)² = k²(a² + b²) = k²c² = (kc)². Aise guna kiya triple scaled roop kehlaata hai, isliye triples anginat hain. 20 tak ki sankhyaon wale triples hain (3, 4, 5), (6, 8, 10), (5, 12, 13), (9, 12, 15), (8, 15, 17) aur (12, 16, 20).

Primitive triples

Jis triple ki sankhyaon ka 1 se bada koi common factor na ho, wo primitive hai. (3, 4, 5), (5, 12, 13) aur (8, 15, 17) primitive hain; (9, 12, 15) nahi, kyunki 3 teeno ko kaatta hai. Kisi non-primitive triple ko uske common factor f se bhaag dene par bhi triple milta hai, kyunki sab kuch f² se bhaag dene par a² + b² = c² bana rehta hai: (9, 12, 15) se (3, 4, 5). Isliye har triple ya to primitive hai ya kisi primitive ka scaled roop, aur saare primitive triples dhoondhna matlab saare triples dhoondhna.

Odd vargon se triples banaana

Odd sankhyaayein kram se jodo: 1 + 3 = 4 = 2², 1 + 3 + 5 = 9 = 3², 1 + 3 + 5 + 7 = 16 = 4². Pehli n odd sankhyaon ka jod hamesha n² hota hai, aur n wi jagah ki odd sankhya 2n − 1 hai. To pehli n − 1 odd sankhyaayein (n − 1)² banaati hain, aur (n − 1)² + (2n − 1) = n². Ab n aisa chuno ki 2n − 1 khud varg ho. 9 = 3², jo 5wi odd sankhya hai, se n = 5 aur 4² + 3² = 5². 25 = 5², jo 13wi hai, se n = 13 aur 12² + 5² = 13². Odd varg 49, 81 aur 121 se (7, 24, 25), (9, 40, 41) aur (11, 60, 61) milte hain.

Aise bane saare triples primitive hote hain, kyunki hypotenuse n aur bhuja n − 1 lagaataar sankhyaayein hain, jinka koi common factor nahi. Par ye tareeka kuch primitive triples chhod deta hai, jaise (8, 15, 17) aur (20, 21, 29), jahan koi bhuja hypotenuse se ek kam nahi.

Cubes ka kya?

Varg anginat tareekon se do vargon ka jod ho sakte hain. 17th century mein French ganitagya Fermat ne daava kiya ki koi cube do cubes ka jod nahi hota, na hi koi badi power do aisi powers ka: n ke 2 se bada hone par xⁿ + yⁿ = zⁿ ka ginti ki sankhyaon mein koi hal nahi. Unka saboot kabhi nahi mila. 300 saal se zyada ki koshishon ke baad Andrew Wiles ne, jinhone 10 saal ki umar mein is sawaal ke baare mein padha tha, 1994 mein ye Fermat’s Last Theorem saabit kiya. Dekho theorem kya hai?, root 2 kya hai?, Chapter 2 ke notes aur NCERT solutions.