Is addition by a specific nonzero natural number a term function in this structure?

Question

Consider the structure $(\mathbb{N};+,\times,0)$. I know that every nonzero natural number $k$ is definable by a first-order formula in that structure, and hence, so is the unary function $x+k$. However, I want to know if the function $x+k$ is a term function in that structure. I strongly suspect it is not, but I want to see the rigorous proof that it is not. I apologize if my question is too elementary or pedantic, but I still want to see the proof.

Answer 1

It is a term function if and only if $k=0$.

Let $t(x)$ be a uni-variate term function. If $t$ has length $1$ then it is either $x$ or $0$. The first case is the term function for $x \mapsto x+0$. Otherwise, $t$ is $t_0 + t_1$ or $t_0 \times t_1$ for term functions $t_0(x), t_1(x)$. In both casses, if $t_0, t_1$ both produce multiples of $x$ then so does $t$. But if $k \neq 0$ then we can take some $x$ such that $x \not\mid x+k$.