output.to from Sideway
Draft for Information Only

# Content

`βPythagorean TriplesβAlgebraic ApproachβHigher DegreeβSource and Reference`

## Pythagorean Triples

Pythagorean Triples: π₯2+π¦2=π§2
If π§β 0, π₯π§2+π¦π§2=1βa unit circle with rational solutions for the homogeneous polynomial.

## Algebraic Approach

β(π)={π+ππ;π,πββ} is a field of number system that build the structure to the rational solutions of π₯2+π¦2=1
βπ₯2+π¦2=(π₯+π¦π)(π₯βπ¦π)=1 β(π₯+π¦π)2(π₯βπ¦π)2=1 rasie to higher power β((π₯2βπ¦2)+(2π₯π¦)π)((π₯2βπ¦2)β(2π₯π¦)π)=1 β(π₯2βπ¦2)2+(2π₯π¦)2=1 structure of the equation is still preserved βπ₯4+π¦4+2π₯2π¦2=1 Similarly, (π₯2+π¦2)2=1βπ₯4+π¦4+2π₯2π¦2=1
Solutions to the equation are
35,45, β725,2425, β5276255,β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+π¦2=(π₯+π¦π)(π₯βπ¦π)=1π§2+π€2=(π§+π€π)(π§βπ€π)=1β((π₯π§βπ¦π€)+(π₯π€+π¦π§)π)((π₯π§βπ¦π€)β(π₯π€+π¦π§)π)=1 β(π₯π§βπ¦π€)2+(π₯π€+π¦π§)2=1 For example, 35,45 and 1213,513β1665,6365 1665,6365 and 35,45ββ204325,253325 That is π={(π₯,π¦)ββ2:π₯2+π¦2=1} π(π₯,π¦),π(π€,π§)βπβπβπβπ, πβπ=(π₯π§βπ¦π€),(π₯π€+π¦π§)) For binary operation:
• πβπ=πβπ
• (πβπ)βπ=πβ(πβπ)
• πΈ=(1,0)βπΈβπ=π

## Higher Degree

Similar to second degree, ```Suppose (π₯+π¦β2+π§β22)(π₯+π¦β2π+π§β22π2)(π₯+π¦β2π2+π§β22π)=1 Let π3=1 and π2+π+1=0, and π4= π3π1=π βπ₯3+2π¦3β6π₯π¦π§+4π§3=1 β{(π₯,π¦,π§)}ββ3:π₯3+2π¦3β6π₯π¦π§+4π§3=1} For Higher Power (π₯+π¦β2+π§β22)n=2,3βπ₯'+π¦'β2+π§'β22 (π₯+π¦β2π+π§β22π2)n=2,3βπ₯'+π¦'β2π+π§'β22π2 ``` By raise the equation to power of 2 (π₯+π¦β2+π§β22)2(π₯+π¦β2π+π§β22π2)2(π₯+π¦β2π2+π§β22π)2=1 β(π₯'+π¦'β2+π§'β22)(π₯'+π¦'β2π+π§'β22π2)(π₯'+π¦'β2π2+π§'β22π)=1 βπ₯'3+2π¦'3β6π₯'π¦'π§'+4π§'3=1 where {π₯'=π₯2+4π¦π§π¦'=π¦2+2π₯π§π§'=2π₯π¦+2π§2

## Source and Reference

ID: 201100015 Last Updated: 11/15/2020 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

Management

HBR 3

Information

Recreation

Culture

Chinese 1097

English 337

Computer

Hardware 153

Software

Application 206

Latex 35

Manim 172

Numeric 19

Programming

Web 285

Unicode 504

HTML 65

CSS 59

ASP.NET 196

OS 422

Python 61

Knowledge

Mathematics

Algebra 31

Geometry 21

Calculus 67

Engineering

Mechanical

Rigid Bodies

Statics 92

Dynamics 37

Control

Natural Sciences

Electric 27