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 Pythagorean Triples
 Geometric Approach
 𝑦-intercept
 Conic Section Example π‘₯2+2𝑦2=1
 Key Ingredients for Binary Operation: π‘₯2+𝑑𝑦2=1
  Generalization to degree 3 with three variables
  Generalization to twists: π‘Žπ‘₯2+𝑏𝑦2=𝑐→π‘₯2+π‘Žπ‘¦2=𝑏
 Key Ingredients for Parameterisation: π‘₯2+𝑑𝑦2=1
 Source and Reference

Pythagorean Triples

image Pythagorean triples: π‘₯2+𝑦2=1 with rational solutions. The rational solution (π‘₯,𝑦) can also be reference to another fixed rational point i.e. (1,0)

Geometric Approach

image By sweeping a line about a fixed rational point (1,0), the slope of the line jointing the rational point (π‘₯,𝑦) can be used to represent this rational point. And the slope is equal to 𝑦π‘₯βˆ’1. Since the rational solutions of π‘₯2+𝑦2=1 always preserves the rationality of slope. In other words, the rational point (π‘₯,𝑦) is mapped to a rational slope. 𝑍0={(π‘₯,𝑦):π‘₯2+𝑦2=1,π‘₯β‰ 1} π‘š:𝑍0β†’β„š given by π‘š(𝑃)=𝑦π‘₯βˆ’1 Similarly, the map π‘š can be defined for more general curves. And the properties of π‘š are
  • π‘š is 1-to-1
  • π‘š is surjective.
  • π‘šβˆ’1:β„šβ†’π‘0 is given by rational functions
These properties⇒a parameterisation

𝑦-intercept

image 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 π‘šβˆ’1(𝑑)=(π‘₯,𝑦); π‘₯2+𝑦2=1 β‡’π‘š(𝑃)=𝑑⇒𝑦π‘₯βˆ’1=𝑑 ⇒𝑦=𝑑(π‘₯βˆ’1) and π‘₯2+𝑦2=1 Let 𝑧=π‘₯βˆ’1 ⇒𝑦=𝑑𝑧 and (𝑧+1)2+𝑦2=1 β‡’(𝑧+1)2+(𝑑𝑧)2=1 ⇒𝑧2+2𝑧+𝑑2𝑧2=0 β‡’(1+𝑑2)𝑧2+2𝑧=0 If 𝑧≠0 β‡’(1+𝑑2)𝑧+𝑧=0 ⇒𝑧=βˆ’21+𝑑2β‡’π‘₯βˆ’1=βˆ’21+𝑑2 β‡’π‘₯=𝑑2βˆ’11+𝑑2βˆŠβ„š ⇒𝑦=𝑑𝑧=π‘‘βˆ’21+𝑑2=βˆ’2𝑑1+𝑑2 β‡’π‘šβˆ’1(𝑑)=(π‘₯,𝑦)=𝑑2βˆ’11+𝑑2,βˆ’2𝑑1+𝑑2

Conic Section Example π‘₯2+2𝑦2=1

Similar to other conic section. image Binary Operation of π‘₯2+2𝑦2=1
Let 𝑃(π‘₯,𝑦),𝑄(𝑀,𝑧) be the solution of the equation. β‡’(π‘₯+π‘¦βˆš2𝑖)(π‘₯βˆ’π‘¦βˆš2𝑖)=1 and (𝑀+π‘§βˆš2𝑖)(π‘€βˆ’π‘§βˆš2𝑖)=1 β‡’(π‘₯π‘§βˆ’2𝑦𝑀)2+2(π‘₯𝑀+𝑦𝑧)2=1 So π‘ƒβŠ•π‘„=𝑅 where 𝑅=(π‘₯π‘§βˆ’2𝑦𝑀,π‘₯𝑀+𝑦𝑧) image map π‘š:𝑍0β†’β„š given by π‘š(𝑃)=𝑦π‘₯βˆ’1 where 𝑍0={(π‘₯,𝑦):π‘₯2+2𝑦2=1, π‘₯β‰ 1} π‘šβˆ’1(𝑑)=(π‘₯,𝑦), where 𝑦=𝑑(π‘₯βˆ’1) Let 𝑧=π‘₯βˆ’1β‡’(𝑧+1)2+2𝑑2𝑧2=1⇒𝑧((1+2𝑑2)𝑧+2)=0 ⇒𝑧=βˆ’2/(1+2𝑑2) β‡’π‘₯=2𝑑2βˆ’11+2𝑑2 ⇒𝑦=βˆ’2𝑑21+2𝑑2 β‡’π‘šβˆ’1(𝑑)=2𝑑2βˆ’11+2𝑑2,βˆ’2𝑑21+2𝑑2 π‘šβˆ’1:β„šβ†’π‘0

Key Ingredients for Binary Operation: π‘₯2+𝑑𝑦2=1

π‘₯2+𝑑𝑦2=1
  • RHS being 1
  • (π‘₯+π‘¦βˆšπ‘‘)𝑛=π‘₯'+𝑦'βˆšπ‘‘β‡’π‘₯2𝑛+𝑑𝑦2𝑛=1

Generalization to degree 3 with three variables

π‘₯3+2𝑦3βˆ’6π‘₯𝑦𝑧+4𝑧3=1β‡’manipulate βˆ›2

Generalization to twists: π‘Žπ‘₯2+𝑏𝑦2=𝑐→π‘₯2+π‘Žπ‘¦2=𝑏

Key Ingredients for Parameterisation: π‘₯2+𝑑𝑦2=1

  • A rational point (1,0)
  • The degree being 2β‡’π‘šβˆ’1:β„šβ†’π‘0

Source and Reference

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

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

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References

  1. B. Joseph, 1978, University Mathematics: A Textbook for Students of Science & 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
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