s-m-n theorem

1. Show that there is a total computable function k such that for each n, k(n) is an index of the function .
2. Show that there is a total computable function k such that for each n, the set of perfect powers.
3. Let . Show that there is a total computable function s such that

   

4. Show that the functions defined in the proof of the s-m-n theorem are primitive recursive.
5. Show that for each m there is a total (m+1)-ary computable function such that for all n

   

   where and are m- and n-tuples respectively.