How to Generate Pythagorean Triples using an Odd Number
1. Start with an odd number, e.g. 3. Then the square of 3 can be expressed as the sum of two consecutive numbers, i.e. 32 = 9 = 4 + 5. Therefore the Pythagorean Triples are {3, 4, 5} since 32 + 42 = 52. Generate another set of Pythagorean Triples starting with the next odd number 5 and verify that it works.
2. Generate another set of Pythagorean Triples starting with the next odd number 7 and verify that it works.
3. Prove that this method will always work for any odd number except 1.
please write exponent with ^ symbol
3^2 = 9
5, ?
25 = sum of 12 and 13
25 + 144 = 169 ?
yes
so 5, 12, 13 works
x^2 = n + (n+1)
is x^2 + n^2 = (n+1)^2 ???
x^2 + n^2 = n^2 +2 n + 1 ???
so if x^2 = 2 n + 1 we have it
but we see 2 n + 1 = n + (n+1)
so true
but what if n is even
try x = 2
x^2 = 4
BUT 4, an even number can not be made from the sum of two consecutive whole numbers. So x must be odd.
To generate Pythagorean triples using an odd number, you can follow these steps:
1. Start with an odd number. Let's take the example of 3.
- Square the odd number. In this case, square 3 to get 9.
- 9 can be expressed as the sum of two consecutive numbers. In this case, 9 = 4 + 5.
- Therefore, the Pythagorean triples are {3, 4, 5} since 3² + 4² = 5².
2. Next, generate another set of Pythagorean triples starting with the next odd number. Let's consider 5.
- Square the odd number. Square 5 to get 25.
- Express 25 as the sum of two consecutive numbers. In this case, 25 = 12 + 13.
- Hence, the Pythagorean triples are {5, 12, 13} since 5² + 12² = 13².
3. Continue this process to generate more Pythagorean triples using subsequent odd numbers (7, 9, 11, and so on) and verify that they work by checking if the sum of the squares of the two smaller numbers equals the square of the largest number.
4. Finally, you can prove that this method will always work for any odd number except 1.
- Let n be an odd number greater than 1.
- The square of n can be expressed as n² = ((n-1)/2)² + ((n+1)/2)². This is based on the fact that (n-1)/2 and (n+1)/2 are consecutive numbers.
- By substituting these values, you can see that n² + ((n-1)/2)² = ((n+1)/2)², which satisfies the Pythagorean theorem.
- Therefore, for any odd number greater than 1, you can always generate Pythagorean triples using this method.
It's worth noting that using an odd number to generate Pythagorean triples gives you primitive triples, which means the three numbers in the triple have no common factors other than 1.