ILL 1988 Problem 63

Let $p$ be the product of two consecutive integers greater than 2. Show that there are no integers $x_1, x_2, \ldots, x_p$ satisfying the equation $$\sum^p_{i = 1} x^2_i - \frac {4}{4 \cdot p + 1} \left( \sum^p_{i = 1} x_i \right)^2 = 1$$

OR

Show that there are only two values of $p$ for which there are integers $x_1, x_2, \ldots, x_p$ satisfying $$\sum^p_{i = 1} x^2_i - \frac {4}{4 \cdot p + 1} \left( \sum^p_{i = 1} x_i \right)^2 = 1$$