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Fibonacci and another sequence

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November 11, 2005
inductionnumber theory unsolvednumber theory

Problem Statement

FnF_n is the Fibonacci sequence F0=F1=1F_0 = F_1 = 1, Fn+2=Fn+1+FnF_{n+2} = F_{n+1} + F_n. Find all pairs m>k≥0m > k \geq 0 such that the sequence x0,x1,x2,...x_0, x_1, x_2, ... defined by x0=FkFmx_0 = \frac{F_k}{F_m} and xn+1=2xn−11−xnx_{n+1} = \frac{2x_n - 1}{1 - x_n} for xn≠1x_n \not = 1, or 11 if xn=1x_n = 1, contains the number 11
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