I want to express the following sentence in first order logic. 


There are naturals numbers that can not be expressed as one natural number raised to the power of another natural number other than one. 


Under normal circumstances this is very simple. I am wondering if the the exponent function involved here can be expressed as uninterpreted function. Can we use some combination of \*,+ as interpreted functions to express exponent function as uninterpreted one ? 



Exponent function as uninterpreted function in first order logic
The function $f(m,n)=m^n$ is primitive recursive, so expressible in
first-order arithmetic: there is a formula in three free variables
$F(m,n,p)$ over the language of first-order arithmetic
which is valid in Peano arithmetic for numerals $m$, $n$ and $p$ iff $p=m^n$.


Logic texts (e.g. Boolos and Jeffrey) will prove that primitive recursive
functions can be expressed in this way, but the general method does
not tend to provide nice formulas for concrete examples like this.


