# Infinite GCD

Compute gcd(2002 + 2, 2002^{2}+2, 2002^{3}+2, …)

- 5 August, 2012 -
- Article, Math Problems -
- Tags : 104 problems, number theory, titu
- 0 Comments

In order to evaluate , we first evaluate . Before we start the Euclidean algorithm, we can simplify this problem considerably by rewriting in terms of

(1)

Substituting from , we now need to find

(2)

From , we know that is at most . Since , all we need to do is prove or disprove

This is an easy proof to make.

(3)

Therefore, all we need to show is that . Notice that so

Therefore, because , then it must be the case that .