@article{Knight2026,
  author    = {{Kevin Knight}},
  title     = {{Collatz high cycles do not exist}},
  journal   = {{Discrete Mathematics}},
  year      = {{2026}},
  volume    = {{349}},
  number    = {{3}},
  pages     = {{114812}},
  doi       = {{https://doi.org/10.1016/j.disc.2025.114812}},
  url       = {{https://www.sciencedirect.com/science/article/pii/S0012365X25004200}},
  abstract  = {{The Collatz function takes odd n to (3n+1)/2 and even n to n/2. Under the iterated Collatz function, every positive integer is conjectured to end up in the trivial cycle 1-2-1. Two types of rational Collatz cycles are of special interest. Consider the set S(k,x) consisting of the smallest members of k-length cycles with x odd terms. The circuit contains the smallest member of S(k,x), while the high cycle contains the largest. It is known that no circuits of positive integers exist (except 1-2-1); this paper shows that there are likewise no high cycles of positive integers.}},
  extra_urls= {{https://hal.science/hal-04261183/document}},
  keywords  = {{Number theory}},
  issn      = {{0012-365X}},
}