$a_{n}=\frac{3n-1}{2n-1}$. Show that no product of the form $a_{n_1} a_{n_2} \dots a_{n_k}$ is a power of $2$. [closed]

Let $$a_{n}=\dfrac{3n-1}{2n-1}.$$

Show that any product of at least $2$ terms of this sequence is not a power of $2$.

In other words:

The equation $$a_{n_1} a_{n_2} \cdots a_{n_k} = 2^a$$ has no solutions where $a, n_1, n_2, \dots, n_k$ are positive natural numbers and $k \geq 2$.

I know $$\dfrac{3n-1}{2n-1}\equiv \dfrac{n-1}{-1}\equiv n-1\pmod 2 .$$

closed as unclear what you're asking by Ivan Neretin, Shailesh, Namaste, Parcly Taxel, user223391 Oct 10 '16 at 22:31

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• What does "show that this sequence any product of a first term not powers of $2$" mean? – lulu Oct 8 '16 at 14:55
• @lulu,Now can you understand? – function sug Oct 8 '16 at 15:02
• I think so, though $a_1=2$ is a counterexample. Are you just excluding that one? – lulu Oct 8 '16 at 15:05
• As a quick remark, no subscript $n$ with $n\equiv 2 \pmod 3$ can occur. That's because, for such $n$, $2n-1\equiv 0\pmod 3$ so the denominator is divisible by $3$, and it is clearly not possible for the numerator to ever cancel that. – lulu Oct 8 '16 at 15:10