document.write( "Question 1030235: Prove that there are no positive integers x, y, z, t such that
\n" ); document.write( "x^2 + y^2 − 7z^2 − 7t^2 = 0.
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #645130 by robertb(5830) $\"\"$ $\"About$

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.)
\n" ); document.write( "But the negative Pell equation $\"y%5E2+-+7t%5E2+=+-1\"$ is not known to be solvable. \r
\n" ); document.write( "\n" ); document.write( "Adding the corresponding sides of these two equations, we get\r
\n" ); document.write( "\n" ); document.write( " $\"x%5E2+%2B+y%5E2+-+7z%5E2+-+7t%5E2+=+0\"$ ,\r
\n" ); document.write( "\n" ); document.write( "which, by the preceding arguments, forces us to conclude that no positive integers x, y, z, t can satisfy the original equation simultaneously.
\n" ); document.write( " \n" ); document.write( "

\n" );