Further Modifications of Original Proofs
My original construction of a Diophantine relation of exponential growth was based on the study of Fibonacchi numbers defined by recurrent relations
while Julia Robinson worked with solutions of the following special kind of Pell equation:
Solutions of this equation (xo, fo), (x1, fa), •••, (xn, fn), ... listed in the order of growth, satisfy the recurrence relations
Sequences ф0,ф,... and ^0,^1,... have many similar properties, for example, they grow up exponentially fast. Immediately after the acquaintance with my construction for Fibonacci numbers, Martin Davis gave in  a Diophantine definition of the sequence of solutions of the Pell equation (2.12). The freedom in selection of the value of the parameter a allowed Martin Davis to construct a Diophantine definition (2.9) of the exponentiation directly, that is, without using the general method proposed by Julia Robinson starting with an arbitrary Diophantine relation of exponential growth. Today the use of the Pell equation for defining the exponentiation by a Diophantine equation has become a standard.