There are various open problems in the subject of logical number theory concerning the possibility of proving this or that well-known standard results over this or that weak theory of arithmetic, usually weakened by restricting the quantifier complexity of the formulas for which one has an induction axiom. It particular, the question of proving the infinitude of the primes in Bounded Arithmetic has received attention.
Does this question make known contact with "workaday number theory" - number theory not informed by concepts from logic and model theory? I understand that proof of the infinitude of the primes in bounded arithmetic could not use any functions that grow exponentially (since the theory doesn't have the power to prove the totality of any such function). So especially I mean to ask:
1) If one had such a proof, would it have consequences about the primes or anything else in the standard model of arithmetic? 2) If one proves that no such proof exists, would that have consequences... 3) Do any purely number theoretic conjectures if settled in the right way, settle this question of its kin?
As a side question, I'd be interested to know the history of this question. I first heard about it from Angus Macintyre and that must have been 25 years ago.