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dc.contributor.authorBatang, Zenon B.
dc.date.accessioned2019-12-24T12:37:49Z
dc.date.available2019-12-24T12:37:49Z
dc.date.issued2019-07-14
dc.identifier.urihttp://hdl.handle.net/10754/660813
dc.description.abstractThe Collatz conjecture asserts that repeatedly iterating $f(x) = (3x + 1)/2^{a(x)}$, where $a(x)$ is the highest exponent for which $2^{a(x)}$ exactly divides $3x+1$, always lead to $1$ for any odd positive integer $x$. Here, we present an arborescence graph constructed from iterations of $g(x) = (2^{e(x)}x - 1)/3$, which is the inverse of $f(x)$ and where $x \not \equiv [0]_3$ and $e(x)$ is any positive integer satisfying $2^{e(x)}x - 1 \equiv [0]_3$, with $[0]_3$ denoting $0\pmod{3}$. The integer patterns inferred from the resulting arborescence provide new insights into proving the validity of the conjecture.
dc.description.sponsorshipCritical remarks from an anonymous reviewer of an earlier version of the manuscript substantially improved the approach presented in this paper. The support from KAUST Core Labs is deeply appreciated.
dc.publisherarXiv
dc.relation.urlhttps://arxiv.org/pdf/1907.07088
dc.rightsArchived with thanks to arXiv
dc.titleInteger patterns in Collatz sequences
dc.typePreprint
dc.contributor.departmentResearch Support
dc.eprint.versionPre-print
dc.identifier.arxivid1907.07088
kaust.personBatang, Zenon B.
refterms.dateFOA2019-12-24T12:38:04Z
kaust.acknowledged.supportUnitCore Labs
kaust.acknowledged.supportUnitKAUST Core Lab


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