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Prove that b divides a(u_b-1)

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September 29, 2010
polynomialnumber theorySequenceDivisibilityrecurrence relationIMO ShortlistIMO Longlist

Problem Statement

(GBR5)(GBR 5) Let us define u0=0,u1=1u_0 = 0, u_1 = 1 and for n≥0,un+2=aun+1+bun,an\ge 0, u_{n+2} = au_{n+1}+bu_n, a and bb being positive integers. Express unu_n as a polynomial in aa and b.b. Prove the result. Given that bb is prime, prove that bb divides a(ub−1).a(u_b -1).
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