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

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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) About Me  (Show Source):
You can put this solution on YOUR website!
The Pell equation x%5E2+-+7z%5E2+=+1 is known to have solutions x = 8 and z = 3. (The smallest known solutions.)
But the negative Pell equation y%5E2+-+7t%5E2+=+-1 is not known to be solvable.
Adding the corresponding sides of these two equations, we get
x%5E2+%2B+y%5E2+-+7z%5E2+-+7t%5E2+=+0,
which, by the preceding arguments, forces us to conclude that no positive integers x, y, z, t can satisfy the original equation simultaneously.