Publication [3.5] of Tomás Oliveira e Silva

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Tomás Oliveira e Silva, "Empirical Verification of the 3x+1 and Related Conjectures." In "The Ultimate Challenge: The 3x+1 Problem," (edited by Jeffrey C. Lagarias), pp. 189-207, American Mathematical Society, 2010.


This paper describes the strategy used by the author to verify empirically the truth of the 3x+1 conjecture, also known as the Collatz conjecture, for all positive integers up to 20\times 2^58\approx 5.7646\times 10^18. It also presents some empirical results concerning two generalized Collatz mappings which resemble in spirit the 3x+1 mapping. For each of these two mappings the growth rate of the so-called maximum excursion record-holders is compared with the conjectured growth rate obtained from the study of a Markov process which was designed to mimic the behavior of the iterates of the generalized mapping.

BibTeX entry

  author = {Oliveira e Silva, Tom{\'a}s},
  title = {Empirical Verification of the $3x+1$ and Related Conjectures},
  booktitle = {The Ultimate Challenge: The $3x+1$ Problem},
  publisher = {American Mathematical Society},
  year = {2010},
  editor = {Lagarias, Jeffrey C.},
  pages = {189--207},
  address = {Providence, Rhode Island, USA}