On the “3𝑥+1” problem

Abstract

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