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a^2 + b^2 not prime, ax (x^2 + ax + 1) = b (x^2 + b + 1) (Chile NMO 2000 P7)

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December 4, 2021
number theoryprime

Problem Statement

Consider the following equation in xx: ax(x2+ax+1)=b(x2+b+1).ax (x^2 + ax + 1) = b (x^2 + b + 1). It is known that a,ba, b are real such that ab<0ab <0 and furthermore the equation has exactly two integer roots positive. Prove that under these conditions a2+b2a^2 + b^2 is not a prime number.
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