MathDB
P-representable Polynomial

Source: IberoAmerican, Day 2, P6

September 9, 2023
algebrapolynomial

Problem Statement

Let PP be a polynomial of degree greater than or equal to 44 with integer coefficients. An integer xx is called PP-representable if there exists integer numbers aa and bb such that x=P(a)−P(b)x = P(a) - P(b). Prove that, if for all N≥0N \geq 0, more than half of the integers of the set {0,1,…,N}\{0,1,\dots,N\} are PP-representable, then all the even integers are PP-representable or all the odd integers are PP-representable.
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