MathDB
Where Fibonacci meets Farey

Source: Kürschák 2002, problem 2

July 8, 2014
number theory unsolvednumber theory

Problem Statement

The Fibonacci sequence is defined as f1=f2=1f_1=f_2=1, fn+2=fn+1+fnf_{n+2}=f_{n+1}+f_n (n∈Nn\in\mathbb{N}). Suppose that aa and bb are positive integers such that ab\frac ab lies between the two fractions fnfn−1\frac{f_n}{f_{n-1}} and fn+1fn\frac{f_{n+1}}{f_{n}}. Show that b≥fn+1b\ge f_{n+1}.
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