Is there an index $i$ such that $\phi_{p(i)}(0) = i + 2$, for a total computable function $p$?
I know about the s-m-n theorem and fixed point theorem, and how to apply them to some basic functions in order to show whether they are computable or not. But, I don't know, how can I find an index of a function as in the exercise above.
I mean, if I have a function like the one below
$$g(x,y) = \begin{cases} 1 & \mbox{if } x = y \\ \uparrow & \mbox{otherwise } \end{cases}$$
where $\uparrow$ means converges, then I can see that the function is computable. We let $t$ be its index, and by the s-m-n theorem we get $$g(x,y) \equiv \phi_{s_1^1(t,x)}(y)$$
$s_1^1(t,x)$ is a function only depending on the argument $x$ while $t$ is fixed, and hence it can be written as $f(x)$. By the fixed point theorem then $$\phi_{f(k)} \simeq \phi_k$$
But, as I said, I don't know how to approach this problem, I would like to see how people approach this kind of problems and solve them. Any ideas?
HINT: Fix some $e$ such that $\varphi_e(0)=0$. Then what can you say about the (total computable) function $$p: i\mapsto e\quad ?$$