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The exercise you’ve uploaded deals with classifying all possible triples of integers \( a \), \( b \), and \( c \) such that:

\[
a^2 + b^2 = c^2.
\]

This is the Pythagorean equation, and the goal is to explore its rational solutions and relate them to points on the unit circle \( S^1 \). Here’s a breakdown of the tasks:

1. **Part (a):** The task is to show that for any rational Pythagorean triple \( (a, b, c) \), the corresponding ratios \( (a/c, b/c) \) lie on the unit circle \( S^1 \). Conversely, for any rational point \( (x, y) \) on \( S^1 \) and a rational number \( c \), we can form a rational Pythagorean triple. This suggests we can classify rational Pythagorean triples by studying rational points on the unit circle.

2. **Part (b):** You are asked to prove that a line \( \ell \) of rational slope \( t \), passing through the point \( (-1, 0) \), intersects the unit circle at exactly one more rational point \( (x, y) \). To do this, you'll set up the equation of the line and use the equation for the unit circle, solving for the second point of intersection.

3. **Part (c):** You need to express the coordinates \( x \) and \( y \) of the rational point on the unit circle in terms of the parameter \( t \), which represents the slope of the line from part (b).

4. **Part (d):** Now, you substitute \( t = r/s \) (a rational number in lowest terms) and show that \( (a, b, c) \) is a Pythagorean triple with the formulas:

   \[
   a = r^2 - s^2, \quad b = 2rs, \quad c = r^2 + s^2.
   \]

   You will also verify that these formulas satisfy the Pythagorean equation.

5. **Part (e):** This part asks you to find the six smallest examples of integer Pythagorean triples.

6. **Part (f):** Lastly, this part involves giving an argument that, for any real number \( \alpha \), there is a rational number \( t \) arbitrarily close to \( \alpha \), which gives rise to a right triangle whose angles are close to \( \theta \), and the sides of the triangle have integer lengths.

Would you like detailed explanations or a step-by-step solution for any particular part of the problem? Let me know if you have specific questions!

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**Relative questions:**

1. How do rational points on the unit circle relate to Pythagorean triples?
2. What is the significance of the rational slope \( t \) in determining Pythagorean triples?
3. How can you verify that the formulas for \( a \), \( b \), and \( c \) satisfy the Pythagorean equation?
4. What is the geometric interpretation of the relationship between lines and circles in this context?
5. What are the smallest Pythagorean triples, and how are they derived?

**Tip:** Rational parametrizations of the unit circle can simplify the classification of Pythagorean triples, transforming complex equations into easier forms!

Classifying all possible triples of integers a, b, and c such that a^2 + b^2 = c^2, including rational parametrization on the unit circle.

Explore how to classify all possible integer Pythagorean triples using rational parametrization on the unit circle. This involves finding rational points on S^1, verifying the relationship to Pythagorean triples, and discovering integer solutions for a, b, and c.

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