The greatest possible value of the GCD of two consecutive Fibonacci numbers is 1.
The greatest possible value of the greatest common divisor (GCD) of two consecutive Fibonacci numbers can be found by considering the properties of Fibonacci numbers.
It is known that for any two consecutive Fibonacci numbers F_n and F_n+1 , the GCD is always 1. This property holds true for all positive integers n.
To prove this, you can use the Euclidean algorithm. Assume d is the GCD of F_n and F_n+1 .
Then, d must also be a divisor of F_n −F_n−1
because:
d∣F_n+1 −F_n
d∣(F_n +F_n−1 )−F_n
d∣F_n−1
This implies that d is also a divisor of F_n and F_n−1 .
Using the same logic, d is also a divisor of F_n−1 and F_n−2 , and so on. Eventually, you reach d being a divisor of F_2 and F_1 , which are both 1. Therefore, d must be 1.
So, the greatest possible value of the GCD of two consecutive Fibonacci numbers is 1.