Pell equations and exponentiation in fragments of arithmetic (Q1919524)

The paper is devoted to the study of the relative strength of two axioms: (1) every Pell equation has a nontrivial solution, and (2) exponentiation is total. It is shown that they are equivalent over \(\text{IE}_1\). Further the graph of the exponential function is defined using only existentially bounded quantifiers in the language of arithmetic expanded with the symbol \#, where \(\# (x, y)= x^{[\log_2 y]}\). The recursion laws of exponentiation are proved in the corresponding fragement.