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Integer polynomial with three fixed values

Source: 2019 Baltic Way P20

November 18, 2019
polynomialnumber theory

Problem Statement

Let us consider a polynomial P(x)P(x) with integers coefficients satisfying P(−1)=−4, P(−3)=−40, and P(−5)=−156.P(-1)=-4,\ P(-3)=-40,\text{ and } P(-5)=-156. What is the largest possible number of integers xx satisfying P(P(x))=x2?P(P(x))=x^2?
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