Pell equations and exponentiation in fragments of arithmetic Original Research Article

We study the relative strength of the two axioms (P) Every Pell equation has a nontrivial solution (exp) 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, y) = x[log2y]. We prove the recursion laws of exponentiation in the corresponding fragment.