$ a , b $ are two odd integers with $\gcd(a,b)=1$. Show that $\gcd(2^a+1,2^b+1)=3$.

Question

$ a , b $ are two odd integers with $\gcd(a,b)=1$. Show that $\gcd(2^a+1,2^b+1)=3$.

We have that $3\mid 2^a+1$ and $3\mid 2^b+1$ because $a$ and $b$ are odd. If we put $d=\gcd(2^a+1,2^b+1)$ I want $3\mid d$ to conclude. Do you have any idea ?

Answer 1

Hint: Consider the Euclidean algorithm (the numbers are allowed to go negative, or equivalently, your allowed to flip the sign of a $\gcd$ term). How much can you reduce the exponents?