We show that Matijasevič's Theorem on the diophantine representation of r.e. predicates is provable in the subsystem I ∃ - 1 of Peano Arithmetic formed by restricting the induction scheme to diophantine formulas with no parameters. More specifically, I ∃ - 1 ⊢ IE - 1 + E ⊢ Matijasevič's Theorem where IE - 1 is the scheme of parameter-free bounded existential induction and E is an ∀∃ axiom expressing the existence of a function of exponential growth. We conclude by (...) examining the consequences of these results to the structure of countable nonstandard models of IE - 1. (shrink)

We prove that the theory IΔ0, extended by a weak version of the Δ0-Pigeonhole Principle, proves that every integer is the sum of four squares (Lagrange's theorem). Since the required weak version is derivable from the theory IΔ0 + ∀x (xlog(x) exists), our results give a positive answer to a question of Macintyre (1986). In the rest of the paper we consider the number-theoretical consequences of a new combinatorial principle, the ‘Δ0-Equipartition Principle’ (Δ0EQ). In particular we give a new proof, (...) which can be formalized in IΔ0 + Δ0EQ, of the fact that every prime of the form 4n + 1 is the sum of two squares. (shrink)