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4 Linear Recurrence Relations & the Fibonacci Sequence

4 Linear Recurrence Relations & the Fibonacci SequenceRecall the classic example of the Fibonacci Sequence (Fn) n=1= (1, 1, 2, 3, 5, 8, 13, 21, ..), defined by{Fn+2=Fn+1+FnF1=F2=1 This Sequence has well-known Relations to population growth (famously breeding rabbits), spirals inthe center of sunflowers, etc. From a number theory perspective, we have two main questions:1. How do we find a formula for thenthFibonacci number? More generally, how do we solvelinear Recurrence Relations ?2. Does the Fibonacci Sequence satisfy any interesting patterns when we consider its remaindersmodulo an integer?}

This defines the sequence in reverse, starting from any pair. In particular, the reverse sequences starting from the pairs (Fn, F n+1) and (F n+N, F n+1+N) in (†) are identical, whence the periodicity continues back to the initial pair (F 1, F 2). Definition 4.6. Denote by N(m) the period of the Fibonacci sequence modulo m; that is, the value

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