# How to show that f(x) is primitive recursive?

Let

$$f(x)=\begin{cases} x \quad \text{if Goldbach's conjecture is true }\\ 0 \quad \text{otherwise}\end{cases}$$

Show that f(x) is primitive recursive.

I know a primitive recursive function is obtained by composition or recursion, but I don't know what should I do about this problem.

• – Raphael Oct 17 '12 at 20:07

Goldbach's conjecture is either true or false. Do a case analysis on the two possibilities. In one case, $f(x)=x$, which is primitive recursive. In the other case, $f(x)=0$, which is also primitive recursive. Therefore $f$ is primitive recursive.