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Maximum excursion and stopping time
record-holders for the $3x+1$ problem: Computational results


Author: Tomás Oliveira e Silva
Journal: Math. Comp. 68 (1999), 371-384
MSC (1991): Primary 26A18; Secondary 11Y99
DOI: https://doi.org/10.1090/S0025-5718-99-01031-5
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Abstract: This paper presents some results concerning the search for initial values to the so-called $3x+1$ problem which give rise either to function iterates that attain a maximum value higher than all function iterates for all smaller initial values, or which have a stopping time higher than those of all smaller initial values. Our computational results suggest that for an initial value of $n$, the maximum value of the function iterates is bounded from above by $n^2 f(n)$, with $f(n)$ either a constant or a very slowly increasing function of $n$. As a by-product of this (exhaustive) search, which was performed up to $n=3 \cdot 2^{53}\approx 2.702 \cdot 10^{16}\!$, the $3x+1$ conjecture was verified up to that same number.


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Additional Information

Tomás Oliveira e Silva
Affiliation: Departamento de Electrónica e Telecomunicações / INESC Aveiro, Universidade de Aveiro, 3810 Aveiro, Portugal
Email: tos@inesca.pt

DOI: https://doi.org/10.1090/S0025-5718-99-01031-5
Keywords: $3x+1$ problem, Collatz problem, algorithm, search, $3x+1$ conjecture
Received by editor(s): January 3, 1997
Article copyright: © Copyright 1999 American Mathematical Society