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Did the mathematician Jeffrey Lagarias prove that in his work the Collatz sequences could not go to infinity (divergent trajectory), that only cyclicity can exist?

I don't have enough mathematics to analyze the book myself. But I think it was implied in the book.

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    What makes you think that way? – Severin Schraven Aug 10 '19 at 10:16
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    No, see this post, which is from $2015$, so later than the survey article by Lagarias. See the comment by Gottfried Helms. If you search this site you will find more posts about it, e.g., here, etc. – Dietrich Burde Aug 10 '19 at 10:17
  • From mathworld: "Conway proved that the original Collatz problem has no nontrivial cycles of length $< 400$. Lagarias ($1985$) showed that there are no nontrivial cycles with length $< 275000$". This result does not mean there might be non-trivial cycles of length with bigger periodicity than these. Not that this has anything to do with your question though. –  Aug 10 '19 at 11:10

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That has not yet been proven.

(And even there was no viable path found yet to do a proof in finite time)

Gottfried Helms
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