Question 2
For a given positive integer $N$, consider
$$x^2+y^2+z^2+t^2-Nxyzt-N=0.$$
(a) Prove that infinitely many values of $N$ admit a solution in positive integers $x,y,z,t$.
(b) If $N=4^k(8m+7)$, where $k,m$ are nonnegative integers, prove that there is no positive integer solution.
$$x^2+y^2+z^2+t^2-Nxyzt-N=0.$$
(a) Prove that infinitely many values of $N$ admit a solution in positive integers $x,y,z,t$.
(b) If $N=4^k(8m+7)$, where $k,m$ are nonnegative integers, prove that there is no positive integer solution.