Talk:Artin's conjecture on primitive roots
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There is still something completely wrong. The density equals Artin's constant when the squarefree part of a is not congruent to 1 modulo 4 (not, as now stated, when a is squarefree).
The reference to Schnirelmann density was wrong. For example, the set S(2) has Schnirelmann density 0 since it doesn't contain p=2. I have written "asymptotic density" instead.
I'm confused about the sense that we're claiming that Heath-Brown's result is nonconstructive. Are we saying that the proof of the statement
- NOT (there exist distinct primes a, b, c such that (NOT ACa) AND (NOT ACb) AND (NOT ACc))
is nonconstructive [where by "ACn" I mean that Artin's conjecture holds for the integer n]? Or are we merely noting that the passage from this statement to
- for all distinct primes a, b, c, ACa OR ACb OR ACc
is nonconstructive? (Or is it that Heath-Brown's proof naturally results in the second statement, having used nonconstructive reasoning somewhere rather earlier, and it's not at all obvious whether the first statement has a constructive proof? Or that Heath-Brown's paper, even if it did prove the first statement constructively, focussed on the second statement as the important result?)
I ask because there is no particular reason why the first statement couldn't have a constructive proof (whereas a constructive proof of the second statement would be a major breakthrough); while the failure to pass from the first to the second is nothing more than the failure of De Morgan's Law in constructive logic (not particularly about Artin's conjecture).
Assuming for the sake of discussion that Heath-Brown does prove the first statement constructively (regardless of whether he set out with that goal), perhaps what we need to do is this:
- First state his result as we have stated it (the constructively valid form);
- Note that the second statement (which we can keep in the current exemplary form) follows;
- Remark that this final step is nonconstructive;
- Note that in fact we don't know of any integer a for which the conjecture holds!
This involves simply swapping (2) and (3) in the current article, and I'd do it myself ... except that I'm worried that our real point might be that (for whatever reason) even the first statement has not been constructively proved. (Then swapping the order as I've suggested would clarify the matter --but the wrong way!)
--Toby Bartels 19:33, 27 February 2006 (UTC)
- I believe that Heath-Brown's proof of the first statement is non-constructive. Basically, he shows that assuming the existence of three primes a,b,c for which Artin's conjecture fails leads to a contradiction.
- Also, the current version states that it suffices to prove Artin's conjecture for prime numbers a; I don't believe this is correct. Even if we knew that 2 and 3 were both primitive roots modulo infinitely many primes (even the "right" density of primes), I don't think there's any way to conclude that 6 is a primitive root modulo infinitely many primes. - Greg Martin —The preceding unsigned comment was added by 18.104.22.168 (talk) 23:01, 10 May 2007 (UTC).
Heath-Brown's result needs rephrasing, I think
The reference to "at most two exceptional primes for which Artin's conjecture fails" is redundant, and therefore confusing. Is there a special kind of prime called an "exceptional" prime? I don't think so.
If not, then the statement would be a lot clearer if it avoided drama and omitted the word "exceptional", reading simply:
Opening para wrong
I think that there is something non-grammatical about the opening para which says "states that a given integer a which is not a perfect square and not −1 is a primitive root modulo infinitely many primes p". I think there should be a word - "of" or "for" or something between modulo and infinitely. However this sentence is so confusing that I can't work out what it should be. -- SGBailey (talk) 14:56, 27 February 2014 (UTC)