MHB Smallest Number in Set A and Proving It Is Not the Only Member

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Let $A$ be the set of all positive integers $a$ such that $2^{2008}+2^a+1$ is a square. Find the smallest number of $A$ and prove that it is not the only member of $A$.
 
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My solution:

Let $$b=2^{2008}+2^a+1$$

i) If $a=1005$, then we may write:

$$b=\left(2^{1004}+1\right)^2$$

ii) If $a=4014$, then we may write:

$$b=\left(2^{2007}+1\right)^2$$
 
Very well done, MarkFL!

Aren't you my smart admin? Hehehe...
 
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