So I've been reading this paper, and I'm trying to understand the proof of the main theorem (I want to use it in different problem, but first I need to fully understand how this kind of proof works). I have some difficulties with:
- if $u+v$, $t_i , t_f$ are random, or they're some specific numbers
- why $q_{ij} $ and $ q'_{ij} $ do exist
- why do $ q_{ij} \neq p_{fg} $, $ q'_{ij} \neq p_{fg} $ hold (it's at the beginning of the proof, first paragraph)
Edit: I added picture of the theorem, and the beginnig of proof.
The value $u+v$ is arbitrary, it just refers to the fact that there are $u$ coverings that satisfy the $r_{ij} \equiv 2^{a_{ij}} - k_i \mod p_{ij}$ equation, and $v$ coverings that satisfy the $r_{ij} \equiv -2^{a_{ij}} - k_j \mod p_{ij}$ equation. Then $t_i$ is the size of the $i$th covering (and $t_f$ is the same for the $f$th covering).
The existence of $q_{ij}$ and $q'_{ij}$ is claimed via reference [5], which is a 1904 paper you can read on JSTOR. I took a quick look at the paper and I think it's a direct consequence of the main theorem in it, but someone more well-versed in number theory can probably explain it (the paper points to two other references for the same claim, but it looks like they're in languages I don't speak).
Likewise, the claim that $q_{ij} \neq p_{fg}$ appears to be by construction of Theorem III from the 1904 paper.