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For an arithmetic progression of the form $a_i = ki, i \in \mathbb{N}$, the question is trivial - if $k$ is a power of 2, then the progression will generate an infinite number of powers of 2, and no powers of 2 for any other case.

I'm having trouble figuring out the more general $a_0 \neq 0 $ case. What form does the progression $a_0 + k\mathbb{N}$ have to take to generate an infinite number of powers of 2? I.e., if an arithmetic progression generates an infinite number of powers of 2, what will the relationship between $a_0$ and $k$ look like?

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Fix a common difference $d$. Take the $d$ arithmetic progressions $A_0,A_1,\ldots, A_{d-1}$ all with common difference $d$ with the first term for $A_i$ chosen as $i$. They partition the set of positive integers. As they are finite in number at least one of them should contain infinitely many powers of 2.

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Wow, that was pretty straightforward. Is there any way to determine which sequences $A_i$ will contain infinitely many powers of 2 for a fixed difference $d$? – xjtian Apr 28 '14 at 0:43
If $i$ and $d$ have some common factor $k>2$, then every element the corresponding A.P. $A_i$ will be divisible by that $k$, and can never be a power of $2$. So choose $i$ such that $\gcd(i,d)=1$ or $2$. This cuts down the search space. – P Vanchinathan Apr 28 '14 at 2:36
One may also allow the gcd to be a power of 2 instead of just 1 or 2. – P Vanchinathan Apr 29 '14 at 4:24

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