I was reading a paper in graph theory "About the number of directed paths in tournaments", the authors used p-adic order to prove a theorem.
I am new to this field, I made some searches and I didn't find what I was looking for.
They wrote: $v_2(3^t-1) \ge t$
I think this is true always (not only in this paper) but I don't know why.
A similar equality that I guess undergo the same property is,
$$v_2(2^{n-5}×7+3(3^{n-5}-1))=\alpha -1$$
Clearly, we must have $\alpha -1\ge n-5$.
I also didn't know why the last inequality is true.
I read on a site that $v_p(a+b)= min\{v_p(a), v_p(b)\}$ under some conditions. I think they used this here with some other property.
Can anyone clear it up please and maybe add a link to where I could find these properties?
This is a picture of the proof and a link to the paper.
https://www.sciencedirect.com/science/article/abs/pii/S0166218X19304226