Bounds for the $3x+1$ problem using difference inequalities
Volume 109 / 2003
                    
                    
                        Acta Arithmetica 109 (2003), 237-258                    
                                        
                        MSC: Primary 11B37; Secondary 26A18, 90C30.                    
                                        
                        DOI: 10.4064/aa109-3-4