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Pythagorean Triples
 Example π‘₯3+2𝑦3=3
 Source and Reference

Pythagorean Triples

Binary operation on π‘₯2+𝑑𝑦2=1β‡’(π‘₯,𝑦)βŠ•(𝑀,𝑧)=(π‘₯π‘§βˆ’π‘‘π‘¦π‘€,π‘₯𝑀+𝑦𝑧) Consider π‘₯2+𝑑𝑦2=1 over a finite field 𝔽𝑝 Theorem: Wherther or not 𝑑 is a square in 𝔽𝑝, there is a solution 𝑃 such that 𝑍 is generated by 𝑃, i.e. 𝑄=π‘ƒβŠ•β‹―βŠ•π‘ƒ for all π‘„βˆŠπ‘
This single generator is similar to the primitive root mod 𝑃
Proof: Consider the field ext 𝐾=𝔽𝑝(βˆšπ‘‘). Denote: [𝑛]𝑃=π‘ƒβŠ•β‹―βŠ•π‘ƒ. Define 𝑓:𝑍→𝐾 by 𝑓(𝑄)=π‘₯+π‘¦βˆšπ‘‘ with rational solution (π‘₯,𝑦). image Existence of a primitive root: π‘”βˆŠπΎ s.t. {𝑔𝑛: 𝑛=0, 1, β‹―}=𝐾*. β–‘ Fact: A subset of finitely many solutions will not generate 𝑍 under the operation. Key ingredients for parameterisation: π‘₯2+𝑑𝑦2=1
  • A rational point (1,0)
  • The degree being 2β‡’π‘šβˆ’1: 𝑄→𝑍0
No easy way to generalize the operation for π‘₯3+𝑦3=1 β‡’(π‘₯+𝑦)(π‘₯+π‘¦πœ)(π‘₯+π‘¦πœ2)=1, where 𝜁3=1 and 𝜁2+𝜁+1=0 β‡’(π‘₯+𝑦)𝑛(π‘₯+π‘¦πœ)𝑛(π‘₯+π‘¦πœ2)𝑛=1 Fact: Let 𝑍={(π‘₯,𝑦):π‘₯3+𝑦3=1} The existence of Ξ¦:𝑄→𝑍 contradicts some genus formula especially some positive. That is the topological invariant to the complex solution of the cubic curveβ‡’no parameterization.
Idea: One rational point and a tangent line β†’ given a rational point.
Intesection to degree 2 such that a cubic curve of degree 3 to degree 2 and the tangent line with

Example π‘₯3+2𝑦3=3

𝑍={π‘₯3+2𝑦3=3}. 𝑃=(1,1)βˆŠπ‘. Let 𝐹(π‘₯,𝑦)=π‘₯3+2𝑦3βˆ’3 tangent line at (π‘Ž,𝑏)=βˆ‚πΉβˆ‚π‘₯(π‘₯βˆ’π‘Ž)+βˆ‚πΉβˆ‚π‘¦(π‘¦βˆ’π‘)=3π‘Ž2(π‘₯βˆ’π‘Ž)+6𝑏2(π‘¦βˆ’π‘)=0 So 3(π‘₯βˆ’1)+6(π‘¦βˆ’1)=0 and π‘₯3+2𝑦3=3 subst. 𝑧=π‘₯βˆ’1, 𝑀=π‘¦βˆ’1 3𝑧+6𝑀=0 and (𝑧+1)3+2(𝑀+1)3=3 6𝑀2(3βˆ’π‘€)=0⇒𝑀=3⇒𝑦=4, π‘₯=βˆ’5 Idea: repeat with (βˆ’5,4) Line: 25(π‘₯+5)+32(π‘¦βˆ’4)=0 and π‘₯3+2𝑦3=3 subst. 𝑧=π‘₯+5, 𝑀=π‘¦βˆ’4 π‘₯=655/253, 𝑦=βˆ’488/253, 253=11*23 Therefore (1,1)β†’(βˆ’5,4)β†’(655/253,βˆ’488/253)
Idea: Two rational points and a secant line→a rational point
βˆ’741253(π‘₯βˆ’1)βˆ’402253(π‘¦βˆ’1)=0 and π‘₯3+2𝑦3=3 subst. 𝑧=π‘₯βˆ’1, 𝑀=π‘¦βˆ’1 741𝑧+402𝑀=0 and (𝑧+1)3+2(𝑀+1)3=3 27732342𝑀315069223+419922𝑀261009+1080𝑀247=0 Therefore 𝑀→0, βˆ’741253(π‘₯βˆ’1)=0⇒𝑀𝑀+741253=0β‡’a factor of cubic equation By long division ⇒𝑀𝑀+74125327732342𝑀15069223+9108061009=0 β‡’π‘₯=2630918269, 𝑦=344918269β‡’π‘₯β‰ˆ1.44, π‘¦β‰ˆ0.18 Tangent line, π‘ƒβŠ•π‘„ and Secant line, 𝑃 ⇒𝑅 Mordell-Weil Theorem: There are a finite subset of solutions that generate all solutions.

Source and Reference

https://www.youtube.com/watch?v=hrp0GdsqLEg
https://www.youtube.com/watch?v=PZlVYEihCh0

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ID: 201100017 Last Updated: 17/11/2020 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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