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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? Linear Recurrence RelationsThe general theory of Linear recurrences is analogous to that of Linear differential Sequence (xn) n=1satisfies alinear Recurrence relation of order r Nif there exista0.}

Computing Legendre symbols and recalling quadratic reciprocity, we see that 5 p = ( 1)5 1 2 p 1 2 p 5 = p 5 = 1 The congruence c2 5 therefore has a solution, which we may assume is odd, for otherwise we could choose the other solution p c. Now define the sequence Jn c …

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