Question

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.

Answer 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.