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

TL;DR

This paper investigates the properties of the sequence a_n=2^{2^n}+d for d>1, focusing on the unboundedness of common divisors among its terms, extending known results about Fermat numbers.

Contribution

It extends the analysis of pairwise coprimality from Fermat numbers to a broader class of sequences with added parameter d.

Findings

Common divisors of distinct terms can be unbounded for certain d

The sequence exhibits different divisibility properties compared to classical Fermat numbers

Results provide new insights into the structure of generalized exponential sequences

Abstract

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 $d>1$