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

[anemone]

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 $A$.

 

[MarkFL]

My solution:

Spoiler

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:

$$b=\left(2^{2007}+1\right)^2$$

 

[anemone]

Very well done, MarkFL!

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