Determine whether there exists a solution for equation
$a^4+b^4+c^4+d^4=(8k+7) 4^{t}(abcd+1)$
where $a,b,c,d$ are positive integers, $k, t$ are non-negative integers.
It seems hard; but that $(8k+7)4^t $ can be written like (1) $x_1^2+x_2^2+x_3^2+x_4^2$.
I think that you can show that for (1 ) exist a $abcd-1$ thus it product is equal to $a^4+b^4+c^4+d^4$.
[Lagrange's squares and the anothers general forms]
How to solve ?