1) There exists some n such that all integers >= n satisfy the desired property.
2) For n = [a specific constant], all integers >= n satisfy the desired property.
These two statements are not the same, but both imply that there are a finite number of counterexamples. The second one is stronger, since we could prove the statement by enumerating all k < n and checking the statement for each such k; if all these checks pass, then the statement is correct.
This strategy does not work for the first strategy since we do not know n, only that such an n exists. In particular, there could be a non-constructive proof that establishes existence of such an n without providing any way to compute such an n.
From the discussion, it does sound like this paper is proving (2), not (1).
I looked up Artin's primitive root conjecture on Wiki because I wasn't familiar with the result. I think you may be referencing the result that there are at most 2 primes for which the conjecture fails, no? If so, I would agree that's not close to a complete proof.
The difference here is that, in principle, since hypergraphs are just finite combinatorial objects, one could, in theory just run a generic coloring algorithm over the finite number of things left out of the proof to obtain the complete result with a bow tied around it.
In the case of Artin's conjecture, you have two difficulties: identifying 1 or 2 candidate primes for which it may fail, and verifying the conjecture for those primes. Even if you can somehow overcome the first difficulty and identify the candidate primes for which it may fail, in the absence of machinery that lets us make such a conclusion for those two primes, we are reduced to calculating an infinite number of discrete logarithms, which, obviously doesn't fly.
So, TL;DR: the difference here is literally the difference between finite and infinite. There are a finite number of hypergraphs left out, which can, in theory be tested using a finite number of operations to verify the conjecture. Not so with Artin's conjecture.
https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primit...
1) There exists some n such that all integers >= n satisfy the desired property.
2) For n = [a specific constant], all integers >= n satisfy the desired property.
I'm not familiar with Artin's conjecture, but from your description, it satisfies (1) but not (2). The reason is that if there are at most 2 such primes, we can take n to be the larger of the two primes plus one. Since all primes are finite, this choice of n is also finite.
I think the key question is whether the paper described in this article proves (1) or (2). From the discussion, it sounds like it proves (2), which is the stronger result.
> Because of the probabilistic elements, their proof only works for large enough hypergraphs — those with more than a specific number of vertices. This approach is common in combinatorics, and mathematicians consider it a nearly complete proof since it only omits a finite number of hypergraphs.