SOLUTION: Prove that there are no positive integers x, y, z, t such that x^2 + y^2 − 7z^2 − 7t^2 = 0.

Question 1030235: Prove that there are no positive integers x, y, z, t such that
x^2 + y^2 − 7z^2 − 7t^2 = 0.

Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
The Pell equation is known to have solutions x = 8 and z = 3. (The smallest known solutions.)
But the negative Pell equation is not known to be solvable.
Adding the corresponding sides of these two equations, we get
,
which, by the preceding arguments, forces us to conclude that no positive integers x, y, z, t can satisfy the original equation simultaneously.

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SOLUTION: Prove that there are no positive integers x, y, z, t such that x^2 + y^2 &#8722; 7z^2 &#8722; 7t^2 = 0.

SOLUTION: Prove that there are no positive integers x, y, z, t such that x^2 + y^2 − 7z^2 − 7t^2 = 0.