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let us assume that $x_1, x_2, x_3, x_4, y_1, y_2, y_3, y_4$ are integer numbers and $m$ is an rational number. i want to choose $m$ such that the following equation is never satisfied :

$$(x_1^2 + x_2^2 + x_3^2 + x_4^2 ) - m( y_1^2 + y_2^2 + y_3^2 + y_4^2 ) =0$$

note that $m$ must belong to rational numbers . whatever $m$ get more close to 1 , it is more appropriate for my problem.

thank you

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  • $\begingroup$ With the high number of unknowns, the solution would be cumbersome. Need this equation or with a smaller number of unknowns? $\endgroup$
    – individ
    Sep 8, 2014 at 12:58
  • $\begingroup$ I think there should be a constraint that all $x_i$'s and $y_i$'s are not $0$ at the same time $\endgroup$
    – AgentS
    Sep 8, 2014 at 13:00
  • $\begingroup$ yes x and y are nonzero numbers . but my problem can never be reduced by smaller number of unknowns. $\endgroup$
    – arman sharififar
    Sep 8, 2014 at 13:03
  • $\begingroup$ i mean x_i's and y_i's can not be all zero at the same time $\endgroup$
    – arman sharififar
    Sep 8, 2014 at 13:04
  • $\begingroup$ For any number $m$ you can write a solution. $\endgroup$
    – individ
    Sep 8, 2014 at 13:15

1 Answer 1

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The Four-Square Theorem states that every natural number is the sum of four squares, in other words, you can solve your equation whenever you can solve $a - mb = 0$ for natural numbers $a$ and $b$. This is possible as long as $m$ is positive (assuming $0$ is not a natural number).

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