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sequence of sum of digits of a value by an integer polynomial

Source: 1987 Polish MO Finals p3

January 20, 2020
polynomialSequencesum of digitsnumber theoryalgebra

Problem Statement

w(x)w(x) is a polynomial with integer coefficients. Let pnp_n be the sum of the digits of the number w(n)w(n). Show that some value must occur infinitely often in the sequence p1,p2,p3,...p_1, p_2, p_3, ... .
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