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A Sequence of Polynomials

Source: 2008 MMO Problem #4

September 13, 2011
algebrapolynomialinductionlogarithmsalgebra unsolved

Problem Statement

The sequence of polynomials (an)(a_n) is defined by a0=0a_0=0, a1=x+2 a_1=x+2 and an=an−1+3an−1an−2+an−2a_n=a_{n-1}+3a_{n-1}a_{n-2} +a_{n-2} for n>1n>1. (a) Show for all positive integers k,mk,m: if kk divides mm then aka_k divides ama_m. (b) Find all positive integers nn such that the sum of the roots of polynomial ana_n is an integer.
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