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Quadratic

Source: Indian RMO 2004 Problem 3

February 28, 2006
quadraticsnumber theorygreatest common divisorrelatively primeinductionnumber theory with sequences

Problem Statement

Let α\alpha and β\beta be the roots of the equation x2+mx−1=0x^2 + mx -1 = 0 where mm is an odd integer. Let λn=αn+βn,n≥0\lambda _n = \alpha ^n + \beta ^n , n \geq 0 Prove that (A) λn\lambda _n is an integer (B) gcd ( λn,λn+1\lambda _n , \lambda_{n+1}) = 1 .
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