We study the relative strength of the two axioms Every Pell equation has a nontrivial solution Exponentiation is total over weak fragments, and we show they are equivalent over IE1. We then define the graph of the exponential function using only existentially bounded quantifiers in the language of arithmetic expanded with the symbol #, where # = x[log2y]. We prove the recursion laws of exponentiation in the corresponding fragment.

A generalization is given of McAloon's result on initial segments ofmodels of GlΔ0, the fragment of Peano Arithmetic where the induction scheme is restricted to formulas with bounded quantifiers.

We prove that if G is a Δ 0 -definable function on the natural numbers and F = Π i = 0 n G , then F is also Δ 0 -definable. Moreover, the inductive properties of F can be proved inside the theory IΔ 0.

We consider the theory and its weak fragments in the language of arithmetic expanded with the functional symbol . We prove that and its weak fragments, down to and , are subject to the Tennenbaum phenomenon with respect to , , and . For the last two theories it is still unknown if they may have nonstandard recursive models in the usual language of arithmetic.

In [4] it is shown that only using exponentiation can one prove the existence of non trivial solutions of Pell equations in IΔ 0 . However, in this paper we will prove that any Pell equation has a non trivial solution modulo m for every m in IΔ 0.

In this paper we prove, modulo Schanuel's Conjecture, that there are algorithms which decide if two exponential polynomials in π are equal in ℝ and if two exponential polynomials in π and i coincide in ℂ.

In [4] the authors studied the residue field of a model M of IΔ0 + Ω1 for the principal ideal generated by a prime p. One of the main results is that M/ has a unique extension of each finite degree. In this paper we are interested in understanding the structure of any quotient field of M, i.e. we will study the quotient M/I for I a maximal ideal of M. We prove that any quotient field of M satisfies the (...) property of having a unique extension of each finite degree. We will use some of Cherlin's ideas from [3], where he studies the ideal theory of non standard algebraic integers. (shrink)