Question

Let \(F_{n}\) and \(F_{n+1}\) be consecutive terms in the Fibonacci sequence. Show that \(\operatorname{ged}\left(F_{n+1}, F_{n}\right)=1\).

Short answer

The gcd of two consecutive Fibonacci numbers is always 1.

Step-by-step solution

Understand the Problem

We need to show that the greatest common divisor (gcd) of two consecutive Fibonacci numbers, \(F_{n+1}\) and \(F_{n}\), is 1. The Fibonacci sequence is defined as \(F_0 = 0\), \(F_1 = 1\), and \(F_n = F_{n-1} + F_{n-2}\) for \(n \geq 2\).

Introduce Important Properties

Recall the property of consecutive Fibonacci numbers: any two consecutive Fibonacci numbers are relatively prime. This means that their gcd is 1. We will prove this by demonstrating that there exist two numbers whose gcd with the second is 1, which extends to the form of (\(F_n, F_{n+1} = F_n + F_{n-1}\)).

Use the Euclidean Algorithm

To find the gcd of two numbers, we use the Euclidean Algorithm. This algorithm states that \(\operatorname{gcd}(a, b) = \operatorname{gcd}(b, a \, \text{mod} \, b)\). Applying to Fibonacci numbers, we need to proof \(\operatorname{gcd}(F_{n+1}, F_{n}) = \operatorname{gcd}(F_{n}, F_{n+1} \, \text{mod} \, F_{n}) = \operatorname{gcd}(F_{n}, F_{n-1})\).

Simplify the Euclidean Algorithm for Fibonacci

In Fibonacci numbers, \(F_{n+1} \, \text{mod} \, F_{n} = F_{n+1} - F_{n} = F_{n-1}\). According to the Euclidean algorithm function, the gcd becomes \(\operatorname{gcd}(F_{n}, F_{n-1})\). We continue reducing like this until we reach a base case like \(\operatorname{gcd}(F_2, F_1) = \operatorname{gcd}(1, 1) = 1\).

Conclude Using the Base Case

We reduce the problem step-by-step using this algorithm until we reach a point where values are easily calculable. Their smallest consecutive forms (\(F_2 = 1, F_1 = 1\)) provide us a base conclusion that if gcd(1,1) = 1 then, at each subsequent reduction, all are similar therefore gcd(\(F_{n+1},F_{n}\)) = 1. This proves they are coprime.

Key concepts

Fibonacci Sequence

The Fibonacci Sequence is a fascinating series of numbers where each term is the sum of the two preceding ones. It begins with 0 and 1, and it is described by the recurrence relation: \[ F_0 = 0,\quad F_1 = 1,\quad \text{and}\quad F_n = F_{n-1} + F_{n-2} \quad \text{for} \quad n \geq 2. \]This sequence appears in various natural phenomena, such as the arrangement of leaves on a stem or the spirals of shells. Each number in a Fibonacci sequence is created by adding the previous two numbers, making it a simple, yet powerful way to model growth patterns. The interesting property of consecutive Fibonacci numbers being relatively prime – meaning their greatest common divisor (gcd) is 1 – presents an intriguing and useful characteristic that has applications in different mathematical proofs and concepts.

Euclidean Algorithm

The Euclidean Algorithm is a classical method for finding the greatest common divisor (gcd) of two numbers. The beauty of this algorithm lies in its simplicity and efficiency. According to it, \\[ \operatorname{gcd}(a, b) = \operatorname{gcd}(b, a \mod b), \]where \(a\mod b\) is the remainder when \(a\) is divided by \(b\).To use the Euclidean Algorithm with Fibonacci numbers, consider two consecutive terms, \(F_{n+1}\) and \(F_n\). By applying the formula, the equation becomes:- \(\operatorname{gcd}(F_{n+1}, F_n) = \operatorname{gcd}(F_n, F_{n+1} \mod F_n) = \operatorname{gcd}(F_n, F_{n+1} - F_n) = \operatorname{gcd}(F_n, F_{n-1}).\)This method is repetitively applied, reducing the problem step by step. Ultimately, it reaches a solvable base case like \(\operatorname{gcd}(1,1) = 1\), affirming the fact that consecutive Fibonacci numbers have a gcd of 1.

Relatively Prime

Two numbers are said to be relatively prime, or coprime, if their greatest common divisor (gcd) is 1. This means they have no common positive integer factors other than 1. The concept of relative primeness plays a vital role in number theory and various mathematical proofs. In the context of the Fibonacci sequence, any two consecutive Fibonacci numbers are always relatively prime. This characteristic is not just a curiosity but provides foundational knowledge useful in fields such as cryptography and computer science. It demonstrates a core property of the Fibonacci series, indicating that no matter how large the numbers grow, their basic arithmetic relationship remains consistent. This consistent pattern is a testament to the simplicity and uniformity witnessed within the Fibonacci sequence.

Mathematical Proof

Mathematical proofs provide a logical and rigorous way to establish truths within mathematics. A proof comprises a series of true statements logically linked to demonstrate a particular conclusion is irrefutable.When it comes to proving that consecutive Fibonacci numbers are relatively prime, the proof can be constructed by utilizing the Euclidean Algorithm. Through repeated application of this efficient method, mathematicians can demonstrate that starting with any two consecutive Fibonacci numbers like \(F_{n+1}\) and \(F_n\), one can simplify to \(\operatorname{gcd}(1,1) = 1\).The beauty of mathematical proofs is seen in their elegance and ability to use fundamental concepts like the Euclidean Algorithm to conclude complex ideas. By building upon smaller verified truths, proofs strengthen our understanding of mathematically intertwined concepts, such as the properties of the Fibonacci numbers in relation to their gcd being 1, ensuring they remain coprime.