Re: Property of an equation

On 18 Apr 2007 05:38:39 -0700, Deep <deepkdeb@xxxxxxxxx> wrote:

On Apr 17, 11:55 pm, quasi <q...@xxxxxxxx> wrote:
On 17 Apr 2007 18:36:40 -0700, Deep <deepk...@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
***
Many thanks.I believe we are coming to a conclusion. Let g/h = m/
n (1.c)
If in (1.c) the condition g = m and h = n always hold then g and h
have no factor in common.

No, this is still not clear. You're saying that g and h have no common
factor if the equation g/h = m/n implies g=m and h=n. That doesn't
make sense to me. For example, certainly g/h = (2g)/(2h).

Before you can talk meaningfully about common factors, you need to
specify the set of all possible factors. If g,h were restricted to a
given ring, then perhaps the concept could be made precise. Even then,
you would have to worry about whether factorization in your ring is
always finite and if it is, whether you have unique factorization.

Again, g and h are real. They may belong to any field.

Can we assume that g and h are positive reals, and that x,y,z are
positive integers?

You stated (1), (2), (3) have no solutions even without the given
conditions. Kindly justify your statement mathematically or otherwise.
***
Your feedback and help are appreciated.

I didn't mean to imply that I could prove that there are no solutions.
My claim that there appeared to be only trivial solutions was based on
looking at numerical data. I asked you if you knew of even one
solution satisfying your conditions. Well, do you?

quasi
.

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