Triples from odd squares
When 2n − 1 is a square, we get a Baudhāyana triple.
A square hiding in the L
We know (n − 1)² + (2n − 1) = n². Look at the odd number 2n − 1.
If that odd number is itself a square number, we have two squares adding to a square. For example, the 5th odd number is 9, and 9 = 3 × 3. So 4² + 3² = 5².
Which other odd number do you think could do the same trick?
What this lesson covers
The idea
When the nth odd number 2n – 1 is itself a square, (n – 1)² + (2n – 1) = n² is a sum of two squares equal to a square, giving a Baudhāyana triple.
A square hiding in the L
We know (n − 1)² + (2n − 1) = n². Look at the odd number 2n − 1.
If that odd number is itself a square number, we have two squares adding to a square. For example, the 5th odd number is 9, and 9 = 3 × 3. So 4² + 3² = 5².
Which other odd number do you think could do the same trick?
- 15
- 25
- 27
Odd squares
Tap an odd number m. The toy finds out which odd number it is, then checks whether it is a square number.
Triples from odd squares
If the nth odd number 2n − 1 is a square s², then (n − 1)² + s² = n². So (s, n − 1, n) is a Baudhāyana triple.
9 = 3² is the 5th odd number: (3, 4, 5). 25 = 5² is the 13th odd number: (5, 12, 13). 49 = 7² is the 25th odd number: (7, 24, 25).
Take care with 1: it is a square, but then n = 1 and n − 1 = 0. There is no triangle, so 1 gives nothing.
Notes
If the nth odd number 2n − 1 is a square s², then (n − 1)² + s² = n² and (s, n − 1, n) is a Baudhāyana triple. 9 gives (3, 4, 5), 25 gives (5, 12, 13), 49 gives (7, 24, 25).
Check yourself
49 is an odd square. It is the nth odd number: 2n − 1 = 49. What is n?
Answer: 25
2n − 1 = 49 gives 2n = 50, so n = 25. The triple is (7, 24, 25).
81 = 9 × 9 is an odd square. It gives a triple (9, n − 1, n). What is the hypotenuse n?
Answer: 41
2n − 1 = 81 gives n = 41. Check: 9² + 40² = 81 + 1600 = 1681 = 41².
Which of these odd numbers gives a Baudhāyana triple this way?
Why does the odd square 1 not give a triple?
- 49 — correct. Yes! 49 = 7 × 7 is an odd square. It gives (7, 24, 25).
- 35. 35 is not a square number: 5 × 5 = 25 and 6 × 6 = 36.
- 45. 45 is not a square number: 6 × 6 = 36 and 7 × 7 = 49.
- Because 1 is an odd number.. Odd numbers are what we use: 9, 25 and 49 are odd too. That is not the reason.
- Because then n = 1, and n − 1 = 0 is not a side of a triangle. — correct. Yes! The triple would be (1, 0, 1). A side cannot be 0.
- Because 1 is not a square number.. 1 = 1 × 1 is a square number. The problem is the side n − 1 = 0.