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?

(In reply to possible solution by armando)

We might also want to know if there are such any pairs with a GCD greater than 2.

If we consider the Fibonacci sequence mod(n), then the period for any given n is {1,3,8,6,20,24,...} known as the Pisano period, A001175 in Sloane. However, that's not the whole story because a period can contain more than one zero, so we need also to check whether two successive zeros can occur close to each other.

It seems that the zeros must be evenly spaced: the spacing is given by A001177, for given n it is {1,3,4,6,5,12,...} 'Fibonacci entry points'. Looking at the two sequences, the division A1175/A1177 looks like an obvious step. The result is A001176, a number that is always 1,2, or 4.

So, we can find numbers such that a pair between F(n) and F(n+m) have a GCD greater than 2, where m is less than the corresponding modulus, but if we require that m be 3 or less, than 2 is the only such GCD.

Edited on March 24, 2016, 12:10 am

Posted by broll on 2016-03-24 00:09:54