@article{10_4064_aa109_3_4,
     author = {Ilia Krasikov and Jeffrey C. Lagarias},
     title = {Bounds for the $3x+1$ problem using difference inequalities},
     journal = {Acta Arithmetica},
     pages = {237--258},
     year = {2003},
     volume = {109},
     number = {3},
     doi = {10.4064/aa109-3-4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4064/aa109-3-4/}
}