Talk:Artin's conjecture on primitive roots
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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)
