non-recursive function 
Give a direct proof that the set $\{x|\Phi_x(1) \downarrow\}$ (which is a set of program numbers that halt on input $1$) is not recursive.

I've got an idea that indirect proof must work. Assuming that the set is recursive and ... But I have no idea how to go on from here.
The course source is Computability, Complexity, and Languages, Second Edition: Fundamentals of Theoretical Computer Science by Davis. So I'm familiar with notation used there and understand it better, if possible. 

Try reduce the problem to Halting problem ,
Halting problem: given M, Turing machine and input x,decide if M(x) halts .
so you want to find a computable functions that take instance of Halting problem and convert it to instance of this problem . 
