On the $‘‘3x+1”$ problem

It is an open conjecture that for any positive odd integer m the function C(m) = (3m + 1)/2e(m) where e(m) is chosen so that C(m) is again an odd integer, satisfies Ce(m) = 1 for some h. Here we show that the number of m < x which satisfy the conjecture is at least x for a positive constant c. A connection between the validity of the conjecture and the diophantine equation 2X -3Y p is established. It is shown that if the conjecture fails due to an occurrence m = Ck(m), then k is greater than 17985. Finally, an analogous "qx + r" problem is settled for certain pairs (q, r) 4 (3, 1).