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Putnam 1973 B3

Source: Putnam 1973

May 25, 2022
Putnamnumber theoryprime numberspolynomial

Problem Statement

Consider an integer p>1p>1 with the property that the polynomial x2−x+px^2 - x + p takes prime values for all integers xx such that 0≤x<p0\leq x <p. Show that there is exactly one triple of integers a,b,ca, b, c satisfying the conditions: b2−4ac=1−4p,    0<a≤c,    −a≤b<a.b^2 -4ac = 1-4p,\;\; 0<a \leq c,\;\; -a\leq b<a.
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