Can this equation be factored down? $\frac{(2^{y}-2) - 2^{y-x}}{2^{y}-1} $

Question

Can this equation be factored down so as to be smaller? Or is this as small as it will go?

$$ \frac{(2^{y}-2) - 2^{y-x}}{2^{y}-1} $$

Answer 1

Let $y=3, x=1$, then $2^y-1=7$ and your expression becomes $\cfrac 27$. Whenever the denominator is a prime $p$ and $y\gt 2, y\gt x \gt 0$ the numerator will be an integer less than $p$. So there can be no general cancellation.

However cancellation will occur in come cases, for example when $y=4, x=1$ we get $\cfrac 6{15}=\cfrac25$.

Answer 2

I think twisting it a bit might yield out some results. If you take 2 common, you can get the following.

$$\dfrac {2[(2^{y-1}-1)-2^{y-x-1}]}{(2^y)-1}$$

This is a bit unclear but if you write it on paper then you can notice, The numerator is in the form of a prime(odd no.)-a no. In powers of two., which should be an odd no. And this divided by another prime should be odd again. But since 2 is common for the equation, it must be a fraction with odd numerator and odd denominator times 2. This might help in your calculation..