Putnam 1973 B3

Consider an integer $p>1$ with the property that the polynomial $x^2 - x + p$ takes prime values for all integers $x$ such that $0\leq x <p$ . Show that there is exactly one triple of integers $a, b, c$ satisfying the conditions: $$b^2 -4ac = 1-4p,\; 0<a \leq c,\; -a\leq b<a.$$