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` Pythagorean Triples Geometric Approach 𝑦-intercept Conic Section Example 𝑥2+2𝑦2=1 Key Ingredients for Binary Operation: 𝑥2+𝑑𝑦2=1  Generalization to degree 3 with three variables  Generalization to twists: 𝑎𝑥2+𝑏𝑦2=𝑐→𝑥2+𝑎𝑦2=𝑏 Key Ingredients for Parameterisation: 𝑥2+𝑑𝑦2=1 Source and Reference`

## 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=𝑡21+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+𝑦2𝑖)(𝑥−𝑦2𝑖)=1 and (𝑤+𝑧2𝑖)(𝑤−𝑧2𝑖)=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+𝑦𝑑)𝑛=𝑥'+𝑦'𝑑⇒𝑥2𝑛+𝑑𝑦2𝑛=1

### Generalization to degree 3 with three variables

𝑥3+2𝑦3−6𝑥𝑦𝑧+4𝑧3=1⇒manipulate 2

## Key Ingredients for Parameterisation: 𝑥2+𝑑𝑦2=1

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

## Source and Reference

ID: 201100016 Last Updated: 16/11/2020 Revision: 0 Ref: References

1. B. Joseph, 1978, University Mathematics: A Textbook for Students of Science &amp; Engineering
2. Wheatstone, C., 1854, On the Formation of Powers from Arithmetical Progressions
3. Stroud, K.A., 2001, Engineering Mathematics
4. Coolidge, J.L., 1949, The Story of The Binomial Theorem  Home 5

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