VN-TST 1994 Problem 2

Consider the equation

$$x^2 + y^2 + z^2 + t^2 - N \cdot x \cdot y \cdot z \cdot t - N = 0$$

where $N$ is a given positive integer.

a) Prove that for an infinite number of values of $N$ , this equation has positive integral solutions (each such solution consists of four positive integers $x, y, z, t$ ),

b) Let $N = 4 \cdot k \cdot (8 \cdot m + 7)$ where $k,m$ are no-negative integers. Prove that the considered equation has no positive integral solutions.