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 Pythagorean Triples
 Algebraic Approach
 Higher Degree
 Source and Reference

Pythagorean Triples

image Pythagorean Triples: π‘₯2+𝑦2=𝑧2
If 𝑧≠0, π‘₯𝑧2+𝑦𝑧2=1β‡’a unit circle with rational solutions for the homogeneous polynomial. image

Algebraic Approach

β„š(𝑖)={π‘Ž+𝑏𝑖;π‘Ž,π‘βˆŠβ„š} is a field of number system that build the structure to the rational solutions of π‘₯2+𝑦2=1
β‡’π‘₯2+𝑦2=(π‘₯+𝑦𝑖)(π‘₯βˆ’π‘¦π‘–)=1 β‡’(π‘₯+𝑦𝑖)2(π‘₯βˆ’π‘¦π‘–)2=1 rasie to higher power β‡’((π‘₯2βˆ’π‘¦2)+(2π‘₯𝑦)𝑖)((π‘₯2βˆ’π‘¦2)βˆ’(2π‘₯𝑦)𝑖)=1 β‡’(π‘₯2βˆ’π‘¦2)2+(2π‘₯𝑦)2=1 structure of the equation is still preserved β‡’π‘₯4+𝑦4+2π‘₯2𝑦2=1 Similarly, (π‘₯2+𝑦2)2=1β‡’π‘₯4+𝑦4+2π‘₯2𝑦2=1
Solutions to the equation are
35,45, βˆ’725,2425, βˆ’5276255,βˆ’336625, 164833390625,354144390625, β‹―
The solutions to the equation can be expanded by combining more than one set of solutions through binary operation. {π‘₯2+𝑦2=(π‘₯+𝑦𝑖)(π‘₯βˆ’π‘¦π‘–)=1𝑧2+𝑀2=(𝑧+𝑀𝑖)(π‘§βˆ’π‘€π‘–)=1β‡’((π‘₯π‘§βˆ’π‘¦π‘€)+(π‘₯𝑀+𝑦𝑧)𝑖)((π‘₯π‘§βˆ’π‘¦π‘€)βˆ’(π‘₯𝑀+𝑦𝑧)𝑖)=1 β‡’(π‘₯π‘§βˆ’π‘¦π‘€)2+(π‘₯𝑀+𝑦𝑧)2=1 For example, 35,45 and 1213,513β‡’1665,6365 1665,6365 and 35,45β‡’βˆ’204325,253325 That is 𝑍={(π‘₯,𝑦)βˆŠβ„š2:π‘₯2+𝑦2=1} 𝑃(π‘₯,𝑦),𝑄(𝑀,𝑧)βˆŠπ‘β‡’π‘ƒβŠ•π‘„βˆŠπ‘, π‘ƒβŠ•π‘„=(π‘₯π‘§βˆ’π‘¦π‘€),(π‘₯𝑀+𝑦𝑧)) For binary operation:
  • π‘ƒβŠ•π‘„=π‘„βŠ•π‘ƒ
  • (π‘ƒβŠ•π‘„)βŠ•π‘…=π‘ƒβŠ•(π‘„βŠ•π‘…)
  • 𝐸=(1,0)β‡’πΈβŠ•π‘ƒ=𝑃

Higher Degree

Similar to second degree, Suppose (π‘₯+π‘¦βˆ›2+π‘§βˆ›22)(π‘₯+π‘¦βˆ›2𝜁+π‘§βˆ›22𝜁2)(π‘₯+π‘¦βˆ›2𝜁2+π‘§βˆ›22𝜁)=1 Let 𝜁3=1 and 𝜁2+𝜁+1=0, and 𝜁4= 𝜁3𝜁1=𝜁 β‡’π‘₯3+2𝑦3βˆ’6π‘₯𝑦𝑧+4𝑧3=1 β‡’{(π‘₯,𝑦,𝑧)}βˆŠβ„š3:π‘₯3+2𝑦3βˆ’6π‘₯𝑦𝑧+4𝑧3=1} For Higher Power (π‘₯+π‘¦βˆ›2+π‘§βˆ›22)n=2,3β‡’π‘₯'+𝑦'βˆ›2+𝑧'βˆ›22 (π‘₯+π‘¦βˆ›2𝜁+π‘§βˆ›22𝜁2)n=2,3β‡’π‘₯'+𝑦'βˆ›2𝜁+𝑧'βˆ›22𝜁2 By raise the equation to power of 2 (π‘₯+π‘¦βˆ›2+π‘§βˆ›22)2(π‘₯+π‘¦βˆ›2𝜁+π‘§βˆ›22𝜁2)2(π‘₯+π‘¦βˆ›2𝜁2+π‘§βˆ›22𝜁)2=1 β‡’(π‘₯'+𝑦'βˆ›2+𝑧'βˆ›22)(π‘₯'+𝑦'βˆ›2𝜁+𝑧'βˆ›22𝜁2)(π‘₯'+𝑦'βˆ›2𝜁2+𝑧'βˆ›22𝜁)=1 β‡’π‘₯'3+2𝑦'3βˆ’6π‘₯'𝑦'𝑧'+4𝑧'3=1 where {π‘₯'=π‘₯2+4𝑦𝑧𝑦'=𝑦2+2π‘₯𝑧𝑧'=2π‘₯𝑦+2𝑧2

Source and Reference

https://www.youtube.com/watch?v=nS6YwdKIIKA
https://www.youtube.com/watch?v=ABr3QisSAWQ

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ID: 201100015 Last Updated: 11/15/2020 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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