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    OlympiadHardAlgebra10–12

    A Polynomial That Cannot Reach Two

    Um Polinómio que Não Chega a Dois

    Let P be a polynomial with integer coefficients and let a, b, c be three distinct integers with P(a) = P(b) = P(c) = 1. Prove that the equation P(x) = 2 has no integer solution.

    For integer coefficients, m - n always divides P(m) - P(n); apply that to a supposed solution three times.

    Solution

    Step 1 of 6

    1. 1.First the tool. If P has integer coefficients then for integers m \ne n each term m^{k} - n^{k} is divisible by m - n, so (m - n) \mid P(m) - P(n).