Re: x^3*(x+2y)-y^3*(y+2x)=z^3


"Mike M" <fairandbalanced999@xxxxxxxxxxx> wrote in message news:VvidnUFzvJchPxDbnZ2dnUVZ_rGinZ2d@xxxxxxxxxxxxxx

"Vincenzo Librandi" <vincenzo.librandweoz@xxxxxxxx> wrote in message news:12404382.1183623907124.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxxxxx
Let x,y,z integers

For which values x, y, z the equality

x^3*(x+2y)-y^3*(y+2x)=z^3

is always verified?

Regards,
Vincenzo Librandi


Oops, it was z^3!



x^3*(x+2y)-y^3*(y+2x)=z^3

x^4 - y^4 + 2xy(x^2 - y^2) = z^2

(x2 - y^2)(x + y)^2 = z^2


Will be satisfied if (x^2 - y^2) = u^2

u^2 + y^2 = x^2


Let u = 2cd ; y = (c^2 - d^2) ; x = (c^2 + d^2) ;

Then z = (x + y)u = 4dc^3


OR


Let y = 2cd ; u = (c^2 - d^2) ; x = (c^2 + d^2) ;

Then z = (x + y)u = (c^2 - d^2)(c + d)^2 = (c - d) (c + d)^3


Mike M











.

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