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ContentβPythagorean Triples
Pythagorean Triples![]() If π§β 0, π₯π§ π¦π§ ![]() Algebraic Approachβ(π)={π+ππ;π,πββ} is a field of number system that build the structure to the rational solutions of π₯2+π¦2=1βπ₯2+π¦2=(π₯+π¦π)(π₯βπ¦π)=1 β(π₯+π¦π)2(π₯βπ¦π)2=1 rasie to higher power β((π₯2βπ¦2)+(2π₯π¦)π)((π₯2βπ¦2)β(2π₯π¦)π)=1 β(π₯2βπ¦2)2+(2π₯π¦)2=1 structure of the equation is still preserved βπ₯4+π¦4+2π₯2π¦2=1 Similarly, (π₯2+π¦2)2=1βπ₯4+π¦4+2π₯2π¦2=1 Solutions to the equation are , β, β, , β― The solutions to the equation can be expanded by combining more than one set of solutions through binary operation. { and β
and β βThat is π={(π₯,π¦)ββ2:π₯2+π¦2=1} π(π₯,π¦),π(π€,π§)βπβπβπβπ, πβπ=(π₯π§βπ¦π€),(π₯π€+π¦π§)) For binary operation:
Higher DegreeSimilar to second degree,Suppose (π₯+π¦β2+π§β22)(π₯+π¦β2π+π§β22π2)(π₯+π¦β2π2+π§β22π)=1
Let π3=1 and π2+π+1=0, and π4= π3π1=π
βπ₯3+2π¦3β6π₯π¦π§+4π§3=1
β{(π₯,π¦,π§)}ββ3:π₯3+2π¦3β6π₯π¦π§+4π§3=1}
For Higher Power
(π₯+π¦β2+π§β22)n=2,3βπ₯'+π¦'β2+π§'β22
(π₯+π¦β2π+π§β22π2)n=2,3βπ₯'+π¦'β2π+π§'β22π2
By raise the equation to power of 2
(π₯+π¦β2+π§β22)2(π₯+π¦β2π+π§β22π2)2(π₯+π¦β2π2+π§β22π)2=1
β(π₯'+π¦'β2+π§'β22)(π₯'+π¦'β2π+π§'β22π2)(π₯'+π¦'β2π2+π§'β22π)=1
βπ₯'3+2π¦'3β6π₯'π¦'π§'+4π§'3=1
where { π₯'=π₯2+4π¦π§π¦'=π¦2+2π₯π§π§'=2π₯π¦+2π§2 Source and Referencehttps://www.youtube.com/watch?v=nS6YwdKIIKAhttps://www.youtube.com/watch?v=ABr3QisSAWQ Β©sideway ID: 201100015 Last Updated: 15/11/2020 Revision: 0 Ref: ![]() References
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