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

On the unboundedness of common divisors of distinct terms of the sequence $a_n=2^{2^n}+d$ for $d>1$