We have a sequence $g_0=3$ and $g_n=2+g_0g_1g_2...g_{n-1}$ for $n\in \mathbb{N}$. How can I prove that $\gcd(g_k,g_n)=1$ if $k<n$? I know that this essentially asking to prove that they are relatively prime, but I am unsure where to begin. Could I show that all of the numbers in this sequence are relatively prime?
Asked
Active
Viewed 33 times
0
-
Can you express explicitly (in terms of $g_i$) what $\gcd(g_k, g_n)$ is? – player3236 Oct 02 '20 at 07:18
-
Hint : Suppose, $g_n$ and some $g_i$ with $i<n$ have a common prime factor and consider that $g_i$ is odd for every $i$. – Peter Oct 02 '20 at 07:40