[obastani]

I don't think that's quite right. There are two possible precise statements for such claims:

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.

[Sniffnoy]

Yup, you came in and said it before I could. Knowing how many counterexamples there could be doesn't help very much if you don't have any bound on their size. pmiller2 seems to assume that there is some way to identify the two candidates, but the problem is that there isn't; the proof is nonconstructive. Nonconstructive proofs are a thing, and it's often important to distinguish whether a given existence proof is constructive or not.