Re: Property of an equation

On 17 Apr 2007 18:36:40 -0700, Deep <deepkdeb@xxxxxxxxx> wrote:

...
refer to (1), (2), (3) below under the given conditions.

X ^1/2 = g(g^4 -10g^2h^2 + 5h^4) (1)

Y^1/2 = h(h^4 -10h^2g^2 + 6g^4) (2)

Z = g^2 + h^2 (3)

Conditions: X and Y are nonsquare integers and Z is odd such that (X,
Y, Z) = 1.
g and 5h have no common factor, h and 5g have no common factor.
( Example: sqrt(3) and 2.sqrt(5) have no factor in common), g and h
are non integers.

These conditions need clarification.

Can we assume that g and h are real?

When you say that g and h are non integers, the concept of "no common
factor" needs to be specified more clearly. The example you gave is
not clear enough from my perspective.

Are g and h assumed to be elements of a (not necessarily the same)
quadratic number field?

If you eliminate g and h from those equations, you get a rather ugly
diophantine equation in x,y,z which appears to have only trivial
integer solutions. That suggests that, even if we disregard the
conditions you imposed on g and h, keeping only your conditions on
x,y,z, your equations are unsatisfiable. Do you know of even one
solution to your system satisfying your conditions on x,y,z?

quasi
.

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