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nmego12345
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Homework Statement
Ok so this isn't really a problem, more like a problem set, I'm not sure if I'm able to understand it yet.
The context is determining all the primitive pythogrean triples
- Letting x = a/c and y = b/c, we see that (x, y) is a point on the unit circle
- Determine conditions on t so that one has a parametrization of all positive rational solutions.
- Let t = u/v where u and v are relatively prime natural numbers and v > u. Then substitute these into the parametrization to obtain
Now, it is tempting, but not valid to conclude that the numerators and denominators on each side are equal. However, one can conclude that there is a positive rational number r such that
Explain why this is the case.
Homework Equations
x^2 + y^2 = 1
If x,y and z are vertices of a triangle and the triangle is right at z then
(xz)^2 + (zy)^2 = (xy)^2
The Attempt at a Solution
This is my attempt at understanding and solving this problem set, correct me if I'm wrong
1. I guess this asks me to get the paramaterization for:
(a/c)^2 + (b/c)^2 = 1
let's insert in
x = a/c, y = b/c
x^2 + y^2 = 1
then it's pretty straightforward
cos(t) = x, sin(t) = y
now I have to verify that if one allows the parameter t to take on only rational solutions, then one obtains a parameterization of all the solutions which are rational numbers.
What does the problem mean by "one obtains a a parameterization of all the rational solutions"?
I think it tells me to find the t values for which x, y are rational. They are
0 + mpi/2, pi/6 + mpi/2, pi/4 + mpi/2, pi/3 + m/pi2, pi/2 + m/pi2,
where m belongs to the set ℕ U {0}
2. This is easy to understand
0 < t < pi/2
Done
3. t = u/v
0 < u/v < pi/2
Now I can't understand how do we get
Another thing is when the problem says "not valid to conclude that the numerators and denominators on each side are equal"
does it means "not valid to conclude that v^2 - u^2 = 2uv"?
Thanks