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Pythagorean Triples

Pythagorean triples: 𝑥²+𝑦²=1 with rational solutions. The rational solution (𝑥,𝑦) can also be reference to another fixed rational point i.e. (1,0)

Geometric Approach

By sweeping a line about a fixed rational point (1,0), the slope of the line jointing the rational point (𝑥,𝑦) can be used to represent this rational point. And the slope is equal to 𝑦𝑥−1. Since the rational solutions of 𝑥²+𝑦²=1 always preserves the rationality of slope. In other words, the rational point (𝑥,𝑦) is mapped to a rational slope. 𝑍0={(𝑥,𝑦):𝑥2+𝑦2=1,𝑥≠1} 𝑚:𝑍0→ℚ given by 𝑚(𝑃)=𝑦𝑥−1 Similarly, the map 𝑚 can be defined for more general curves. And the properties of 𝑚 are

• 𝑚 is 1-to-1
• 𝑚 is surjective.
• 𝑚⁻¹:ℚ→𝑍₀ is given by rational functions

These properties⇒a parameterisation

𝑦-intercept

Similar to Riemann stereographic projection, the slope of the sweeping line can be projected as the 𝑦-intercept on the 𝑦-axis. The parameterisation of the 2-degee equation with one fixed point gives one unique association with the remaining point. Let 𝑚⁻¹(𝑡)=(𝑥,𝑦); 𝑥²+𝑦²=1 ⇒𝑚(𝑃)=𝑡⇒𝑦𝑥−1=𝑡 ⇒𝑦=𝑡(𝑥−1) and 𝑥²+𝑦²=1 Let 𝑧=𝑥−1 ⇒𝑦=𝑡𝑧 and (𝑧+1)²+𝑦²=1 ⇒(𝑧+1)²+(𝑡𝑧)²=1 ⇒𝑧²+2𝑧+𝑡²𝑧²=0 ⇒(1+𝑡²)𝑧²+2𝑧=0 If 𝑧≠0 ⇒(1+𝑡²)𝑧+𝑧=0 ⇒𝑧=−21+𝑡²⇒𝑥−1=−21+𝑡² ⇒𝑥=𝑡²−11+𝑡²∊ℚ ⇒𝑦=𝑡𝑧=𝑡−21+𝑡²=−2𝑡1+𝑡² ⇒𝑚⁻¹(𝑡)=(𝑥,𝑦)=𝑡²−11+𝑡²,−2𝑡1+𝑡²

Conic Section Example 𝑥²+2𝑦²=1

Similar to other conic section. Binary Operation of 𝑥²+2𝑦²=1
Let 𝑃(𝑥,𝑦),𝑄(𝑤,𝑧) be the solution of the equation. ⇒(𝑥+𝑦√2𝑖)(𝑥−𝑦√2𝑖)=1 and (𝑤+𝑧√2𝑖)(𝑤−𝑧√2𝑖)=1 ⇒(𝑥𝑧−2𝑦𝑤)²+2(𝑥𝑤+𝑦𝑧)²=1 So 𝑃⊕𝑄=𝑅 where 𝑅=(𝑥𝑧−2𝑦𝑤,𝑥𝑤+𝑦𝑧) map 𝑚:𝑍₀→ℚ given by 𝑚(𝑃)=𝑦𝑥−1 where 𝑍₀={(𝑥,𝑦):𝑥²+2𝑦²=1, 𝑥≠1} 𝑚⁻¹(𝑡)=(𝑥,𝑦), where 𝑦=𝑡(𝑥−1) Let 𝑧=𝑥−1⇒(𝑧+1)²+2𝑡²𝑧²=1⇒𝑧((1+2𝑡²)𝑧+2)=0 ⇒𝑧=−2/(1+2𝑡²) ⇒𝑥=2𝑡²−11+2𝑡² ⇒𝑦=−2𝑡²1+2𝑡² ⇒𝑚⁻¹(𝑡)=2𝑡²−11+2𝑡²,−2𝑡²1+2𝑡² 𝑚⁻¹:ℚ→𝑍₀

Key Ingredients for Binary Operation: 𝑥²+𝑑𝑦²=1

𝑥²+𝑑𝑦²=1

• RHS being 1
• (𝑥+𝑦√𝑑)^𝑛=𝑥'+𝑦'√𝑑⇒𝑥²_𝑛+𝑑𝑦²_𝑛=1

Generalization to degree 3 with three variables

𝑥³+2𝑦³−6𝑥𝑦𝑧+4𝑧³=1⇒manipulate ∛2

Generalization to twists: 𝑎𝑥²+𝑏𝑦²=𝑐→𝑥²+𝑎𝑦²=𝑏

Key Ingredients for Parameterisation: 𝑥²+𝑑𝑦²=1

• A rational point (1,0)
• The degree being 2⇒𝑚⁻¹:ℚ→𝑍₀

Source and Reference

https://www.youtube.com/watch?v=XiwGK8sdwpQ
https://www.youtube.com/watch?v=eLoQz1WRFu0