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Find the greatest common divisor - ILL 1990 MOR2

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September 18, 2010
inductionalgebrapolynomialnumber theorygreatest common divisornumber theory proposed

Problem Statement

The sequence {un}\{u_n\} is defined by u1=1,u2=1,un=un−1+2un−2forn≥3u_1 = 1, u_2 = 1, u_n = u_{n-1} + 2u_{n-2} for n \geq 3. Prove that for any positive integers n,p (p>1),un+p=un+1up+2unup−1n, p \ (p > 1), u_{n+p} = u_{n+1}u_{p} + 2u_nu_{p-1}. Also find the greatest common divisor of unu_n and un+3.u_{n+3}.
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