The Collatz Conjecture

Consider the two functions on integers f3 and f5, such that

• f3 x = x/2 when x is even
• f3 x = 3x+1 otherwise
• f5 x = x/2 when x is even
• f5 x = 5x+1 otherwise

Could it be that

• The function that takes as input a number n and outputs the number of steps f3(n) takes to halt is primitive recursive, and
• The function that takes as input a number n and outputs whether f5(n) diverges is general recursive?