On the unboundedness of common divisors of distinct terms of the sequence a_n=2^(2^n)+d for d>1

It is well-known that for any distinct positive integers $k$ and $n$, the numbers $2^{2^k}+1$ and $2^{2^n}+1$ are relatively prime. In this paper we consider the situation when 1 is replaced by some positive integer $d>1$