This is an extension of
GCD of Fibonacci.
Denote the nth term of the Fibonacci sequence as F(n), with F(0)=0 and F(1)=1. Let S_n be the set {F(n), F(n+1), F(n+2), F(n+3)}.
For what values n does there exist a pair of numbers from S_n with a GCD greater than 1?
Wikipedia says that two consecutives terms of a Fibonacci series are coprimes.
So:
F(n) and F(n+1) are coprimes.
But the sum of two coprimes numbers is coprime with both. Then F(n+2), F(n+1), F(n) are relatively primes
Then F(n+3)=F(n+2)+Fn(n+1)=F(n)+2F(n+1). F(n) and F(n+1) are coprimes but if F(n) is even then F(n+3) will be even.
So, if F(n) is even F(n+3) is even and GDC is 2.
This is confirmed by a look to the Fibonacci sequence which is odd,odd,even,odd,odd,even...
Edited on March 23, 2016, 10:05 am

Posted by armando
on 20160323 09:53:30 