Direct Proofs Using Fibonacci Numbers

Proofs: Number Theory and Sequences

In this course, Professor Shabnam Akhtari (University of Oregon) explores proofs, specifically those motivated by number theory and sequences. In the first mini-lecture, we introduce the Fibonacci numbers and use them to motivate examples of direct proofs. In the second mini-lecture, we introduce proof by contradiction and use this method to prove properties about prime numbers, including Euclid’s Theorem, which states that there are infinitely many prime numbers. In the third mini-lecture, we discuss prime factorisation notation using exponents and work through two related proofs. In the fourth mini-lecture, we consider direct proofs involving arithmetic and geometric sequences.

Direct Proofs Using Fibonacci Numbers

In this mini-lecture, we use the Fibonacci numbers to motivate examples of direct proofs. As we move though this mini-lecture, we consider: (i) the definition of Fibonacci numbers, F_n = F_n-1 + F_n-2, which can be used to create a sequence of numbers; (ii) mathematical patterns in the Fibonacci sequence, such as every third number being even while all others are odd; (iii) a direct proof of a property of Fibonacci numbers, which states that F_n+1 = 2F_n-1 + F_n-2; and (iv) how this property, F_n+1 = 2F_n-1 + F_n-2, can also help us deduce which terms are even and which are odd in the Fibonacci sequence.