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I have recently read the famous paper by Crandall (1978) on the $3x+1$ problem, and I wonder what progress has been made since then.

The paper claims that:

  1. If a cycle exists, then the minimum number of $3x+1$ steps for the number to reach itself is $17985$. Is there a better lower bound nowadays?
  2. It is not known whether a positive density of odd integers satisfy the conjecture. Is that still unknown?
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    $\begingroup$ Terrence Tao has come incredibly close to proving it without proving it. Here is his recent paper. arxiv.org/abs/1909.03562 $\endgroup$ Commented Jan 12, 2022 at 15:58
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    $\begingroup$ Cycle length is known to be at least 114 billion -- reference. $\endgroup$ Commented Jan 12, 2022 at 16:00
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    $\begingroup$ @EricTowers So the lower bound is about $10^{11}$. Did I interprete "billion" correctly ? $\endgroup$
    – Peter
    Commented Jan 12, 2022 at 16:18
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    $\begingroup$ That with the density should be known. It is known that the counterexamples (if there are any) form an extremely tiny set. $\endgroup$
    – Peter
    Commented Jan 12, 2022 at 16:20
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    $\begingroup$ In fact, Barin reports that Collatz is confirmed up to $2^{69}$ (about a month ago). My read of the 1993 paper is that this doesn't move the lower bound (but I skimmed very quickly, so could have missed something important). $\endgroup$ Commented Jan 12, 2022 at 17:03

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As of the date of this post the conjecture has been confirmed to be true up to $$682 \times 2^{60}\left(\approx 2^{69.41}\right).$$

In terms of proving it Terence Tao published a paper titled "Almost all orbits of the Collatz map attain almost bounded values."

"Almost all"! How exciting is that! But who knows, maybe we are still further than we think...

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