# Prove $\prod_{i=1}^n \frac{3K_i+2}{2K_i+1}$ can never be a power of $2$

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?

• We must stipulate $n\ge1$, since an empty product is $2^0$.
– J.G.
Commented Jan 2, 2021 at 10:19

Its not true. For instance, define $$\mathcal K:=\{5,6,8,24,27,32,41,47,69,92\}$$. Then we have, $$\prod_{i\in\mathcal K}\frac{3i+2}{2i+1}=2^6.$$