■整数三角形の話(その10)

 a^2−ab+b^2=c^2一般解は,

c=m^2+mn+n^2,a=m^2+2mn,b=m^2−n^2

または

c=m^2+mn+n^2,b=m^2+2mn,a=2mn+n^2

の形になる.

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 一方,a^2+ab+b^2=c^2の一般解は

a=(m^2−n^2),b=(2mn+n^2),c=(m^2+mn+n^2)

  a^2+ab+b^2=m^4−2m^2n^2+n^4+2m^3n+m^2n^2−2mn^3−n^4+4m^2n^2+4mn^3+n^4

=m^4+2m^3n+3m^2n^2+2mn^3+n^4

=(m^2+mn+n^2)^2=c^2

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