Are there primitive triples that cannot be obtained through this method? If yes, give examples.
Yes, there are primitive Baudhāyana triples that cannot be obtained through the odd square method.
Examples: (8, 15, 17), (20, 21, 29), and (12, 35, 37)
Reason: In these triples, neither of the smaller sides is one less than the hypotenuse.
For (8, 15, 17): Smaller sides: 8 and 15 Hypotenuse: 17 Neither 8 nor 15 equals 16 (one less than 17) ∴ These triples cannot be obtained through the odd square method.
Explanation
The chapter shows that the odd square method generates triples where one smaller side is one less than the hypotenuse. However, Baudhāyana himself listed triples like (8, 15, 17) and (12, 35, 37) which are primitive but don't follow this pattern. In (8, 15, 17), we have 8² + 15² = 64 + 225 = 289 = 17², but neither 8 nor 15 equals 16. These triples are primitive (no common factor) but cannot be generated by the odd square method described in the chapter.
Solution Steps
Identify that primitive Baudhāyana triples exist that cannot be obtained through the odd square method.
List examples: (8, 15, 17), (20, 21, 29), and (12, 35, 37).
Note that in these triples, neither smaller side is one less than the hypotenuse.
Verify: In (8, 15, 17), smaller sides are 8 and 15, hypotenuse is 17, neither equals 16.