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Chapter 0: Section 0.1 - Part 1

1.  "It is not difficult to find a formula for Fn" (1)  The standard recurrence relation for Fibonnaci number n is: F0=0 F1=1 For n≥2 Fn=Fn−1+Fn−2 (2)  The formula for n is as follows: Fn=1√5((1+√52)n−(1−√52)n) Notes:   The formula is known as Binet's Formula  and its derivation showcases the deep relationship between the Fibonacci number and the golden ratio . The derivation and elementary proofs are detailed here . Exercises: 1.  Exercise 0.1.1:  (a)   Use the recurrence relation for the fibonacci numbers, and induction to prove that every Fibonacci number is an integer. Solution: S(1) is true since F1=1 is an integer. We assume that S(k) is true, that is, S(k) is an integer. S(k+1) = S(k) + S(k-1) which is an integer since the sum of two integers is an integer by the closure property on addition.  See here ...
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Notes on Andrew Granville's Number Theory: A Masterclass

I was very excited when I purchased Andrew Granville's Number Theory: A Masterclass .   I had read his graphic novel Prime Suspects and felt overwhelmed.  The main insight presented in the graphic novel, a surprisingly relationship between two very different mathematical ideas, was beyond me.  I encourage anyone interested to purchase the book and give it a try.  I figured that reading through the masterclass would not only deepen my understanding of number theory but also give me a better context for rereading  Prime Suspects . As I read the preface and skimmed through the chapters, I realized that the book was more well-thought out and better organized than I had expected.  I was very impressed that he had taken Gauss's Disquisitiones Arithemticae as the inspiration for his effort.  I am especially excited that he has planned a future book that will provide a walkthrough of Gauss's classic work. To motivate myself to go through the book slowly...
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