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Suppose we have a finite set of numbers $K_i \in \mathbb N$.

Can we prove that the product $$\prod_{i=1}^n \frac{3K_i+2}{2K_i+1}$$

can never be equal a power of $2$?

As you can see, each term is between $\frac{3}{2}$ and $\frac{5}{3}$. The greater $K_i$, the closer it is to $\frac{3}{2}$. Ratio between exact power or $\frac{3}{2}$ and the next greater power of $2$ places the constraint on $K_i$ values: the lowest $K_i$ cannot exceed certain value. For example, for products up to $2000$ terms, the biggest lowest value of $K_i$ was found to be for $1636$ terms, and it is $291643$.

The value would be a power of $2$ if all factors of the numerator and the denominator, except of $2$s in the numerator, completely canceled each other. Note that the denominator is always odd. It appears that there's no set of $K_i$ to make such a product equal to a power of $2$.

But can it be proven?

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  • $\begingroup$ We must stipulate $n\ge1$, since an empty product is $2^0$. $\endgroup$ – J.G. Jan 2 at 10:19
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Its not true. For instance, define $\mathcal K:=\{5,6,8,24,27,32,41,47,69,92\}$. Then we have, $$\prod_{i\in\mathcal K}\frac{3i+2}{2i+1}=2^6.$$

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    $\begingroup$ Thank you, that's a nice counter-example. $\endgroup$ – Alexander Grigoriev Jan 2 at 15:13
  • $\begingroup$ How did you obtain this counter example? $\endgroup$ – N.S.JOHN Jan 24 at 4:21

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