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From: david on 6 Jul 2010 14:15
Consider the following two equations under the given conditions.

x^k + y^k = z^2 (1) x^k + (1/4)y^k =
z^2 (2)

Conditions: x, y, z are pairwise coprimes, 2|y, k|z, prime k > 3, each
variable > 1

Conjecture: If (1) has no integer solutions then (2) also cannot haved
any integer solutions.

Any appropriate reference which will help justify the conjecture will
be gratefully appreciated.

Thanks

David
From: William Elliot on 7 Jul 2010 02:22
On Tue, 6 Jul 2010, david wrote:

> Consider the following two equations under the given conditions.
>
> x^k + y^k = z^2 (1)
> x^k + (1/4)y^k = z^2 (2)
>
> Conditions: x, y, z are pairwise coprimes, 2|y, k|z, prime k > 3, each
> variable > 1
>
> Conjecture: If (1) has no integer solutions then (2) also cannot haved
> any integer solutions.

What solutions are known for (2)?
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