Proving Infinitude of Prime Numbers Using Binomial Coefficients

We study the problem of proving in weak theories of Bounded Arithmetic the theorem that there are arbitrarily large prime numbers. We show that the theorem can be proved by some “minimal” reasoning (i.e., in the theory I∆0) using concepts such as (the logarithm) of a binomial coefficient. In fact we prove Bertrand’s Postulate (that there is at least a prime number between n and 2n, for all n > 1) and the fact that the number of prime numbers between n and 2n is of order Θ(n/ ln(n)). The proofs that we formalize are much simpler than several existing formalizations, and our theory turns out to be a sub-theory of a recent theory proposed by Woods and Cornaros that extends I∆0 by a special counting function.