@article{Simons2007,
  author    = {{John L. Simons}},
  title     = {{A simple (inductive) proof for  the non-existence of $2$-cycles for the $3x+1$ problem}},
  journal   = {{J. Number Theory}},
  year      = {{2007}},
  volume    = {{123}},
  pages     = {{10-17}},
  url       = {{https://www.sciencedirect.com/science/article/pii/S0022314X06001223}},
  abstract  = {{This article generalizes a proof of Steiner for the nonexistence of 1-cycles for the 3x + 1 problem to a proof for the nonexistence of 2-cycles. A lower bound for the cycle length is derived by approximating the ratio between numbers in a cycle. An upper bound is found by applying a result of Laurent, Mignotte, and Nesterenko on linear forms in logarithms. Finally numerical calculation of convergents of log2 3 shows that 2-cycles cannot exist.}},
  extra_urls= {{https://www.researchgate.net/publication/220576261_On_the_nonexistence_of_2-cycles_for_the_3x1_problem}},
}