Over 25 years ago, the first author conjectured in  that the existence of arbitrarily large primes is provable from the axioms IΔ0(π)+def(π), where π(x) is the number of primes not exceeding x, IΔ0(π) denotes the theory of Δ0 induction for the language of arithmetic including the new function symbol π, and def(π) is an axiom expressing the usual recursive definition of π. We prove a modified version in which π is replaced by a more general function ξ that counts some of the primes below x (which primes depends on the values of parameters in ξ), and has the property that π is provably Δ0(ξ) definable.
"On bounded arithmetic augmented by the ability to count certain sets of primes." J. Symbolic Logic 74 (2) 455 - 473, June 2009. https://doi.org/10.2178/jsl/1243948322