Technique explanation
Pythagorean triples (a, b, c) satisfying a² + b² = c² can be generated from any two positive integers m and n, where m > n. The formula derived from the sutra Puranapuranabhyam is: a = m² - n², b = 2mn, and c = m² + n².
This method applies to all positive integers. If m and n are coprime and have opposite parity, the generated triple is primitive. Standard fallback: Geometric construction or reference to known triple lists.
Worked examples
Example 1: Seed numbers 5 and 2
1. Let m=5, n=2.
2. a = 5² - 2² = 21.
3. b = 2(5)(2) = 20.
4. c = 5² + 2² = 29.
5. Triple: (21, 20, 29).
Check: 21² + 20² = 441 + 400 = 841, and 29² = 841. Verified.
Example 2: Seed numbers 4 and 3
1. Let m=4, n=3.
2. a = 4² - 3² = 7.
3. b = 2(4)(3) = 24.
4. c = 4² + 3² = 25.
5. Triple: (7, 24, 25).
Check: 7² + 24² = 49 + 576 = 625, and 25² = 625. Verified.
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