# Publication [3.5] of Tomás Oliveira e Silva

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## Reference

**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.

## Abstract

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

@InCollection { Oliveira.e.Silva-2010-1-EVCC, 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} }