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f = g^2 if f = aX^2 + bX+ c is a perfect square for every n

Source: 2011 Saudi Arabia Pre-TST February 2.3

January 1, 2022
algebrapolynomialPerfect Square

Problem Statement

Let f=aX2+bX+c∈Z[X]f = aX^2 + bX+ c \in Z[X] be a polynomial such that for every positive integer nn,f(n) f(n ) is a perfect square. Prove that f=g2f = g^2 for some polynomial g∈Z[X]g \in Z[X].
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