Abstract
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
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DOI 10.1016/0168-0072(95)00018-6
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References found in this work BETA

On the Scheme of Induction for Bounded Arithmetic Formulas.A. J. Wilkie & J. B. Paris - 1987 - Annals of Pure and Applied Logic 35 (3):261-302.
Existence and Feasibility in Arithmetic.Rohit Parikh - 1971 - Journal of Symbolic Logic 36 (3):494-508.
Diophantine Induction.Richard Kaye - 1990 - Annals of Pure and Applied Logic 46 (1):1-40.
Bounded Existential Induction.George Wilmers - 1985 - Journal of Symbolic Logic 50 (1):72-90.

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Citations of this work BETA

Solving Pell Equations Locally in Models of IΔ0.Paola D'Aquino - 1998 - Journal of Symbolic Logic 63 (2):402-410.
Quadratic Forms in Models of IΔ0+ Ω1. I.Paola D’Aquino & Angus Macintyre - 2007 - Annals of Pure and Applied Logic 148 (1):31-48.
Quadratic Forms in Models of I Δ 0 + Ω 1. I.Paola D’Aquino & Angus Macintyre - 2007 - Annals of Pure and Applied Logic 148 (1-3):31-48.

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