<story><title>Terence Tao Proves Result on the Collatz Conjecture</title><url>https://www.quantamagazine.org/mathematician-terence-tao-and-the-collatz-conjecture-20191211/</url></story><parent_chain><item><author>wil93</author><text>&amp;gt; Tao used this weighting technique to prove that almost all Collatz starting values — 99% or more — eventually reach a value that is quite close to 1. This allowed him to draw conclusions along the lines of 99% of starting values greater than 1 quadrillion eventually reach a value below 200.&lt;p&gt;Isn&amp;#x27;t this equivalent to saying that &amp;quot;99% of starting values greater than 1 quadrillion eventually reach 1&amp;quot;?&lt;p&gt;I mean, once you reach a value below 200 then you will continue and reach 1. Not only below 200, but below any limit that was experimentally verified (i.e. around 10^20)</text></item></parent_chain><comment><author>soVeryTired</author><text>If Colmin(N) is the lowest value reached by iterating the process from initial value N, Tao proved that&lt;p&gt;Colmin(N) &amp;lt; f(N)&lt;p&gt;for &lt;i&gt;arbitrary&lt;/i&gt; f provided f tends to infinity, for almost all N. Here, &amp;quot;almost all&amp;quot; means something like &amp;quot;exceptions(N) &amp;#x2F; N, tends to 0 for large N&amp;quot;, where exceptions(N) is the count of values that do not obey the inequality [0]&lt;p&gt;So it&amp;#x27;s an asymptotic result. The first million integers could all be exceptions - but &lt;i&gt;eventually&lt;/i&gt; the proportion of exceptions dies out.&lt;p&gt;The ability to pick arbitrary f is very powerful. Pick the slowest-growing function you can think of. e.g. Tenfold-iterated logarithm. The inequality says for all but a negligible fraction of integers, Colmin grows slower than that function.&lt;p&gt;[0] Nitpick: I&amp;#x27;m describing the &lt;i&gt;natural&lt;/i&gt; density, but Tao needs the &lt;i&gt;logarithmic&lt;/i&gt; density, where each exception &lt;i&gt;n&lt;/i&gt; is weighted by &lt;i&gt;1&amp;#x2F;n&lt;/i&gt;.</text></comment>
<story><title>Terence Tao Proves Result on the Collatz Conjecture</title><url>https://www.quantamagazine.org/mathematician-terence-tao-and-the-collatz-conjecture-20191211/</url></story><parent_chain><item><author>wil93</author><text>&amp;gt; Tao used this weighting technique to prove that almost all Collatz starting values — 99% or more — eventually reach a value that is quite close to 1. This allowed him to draw conclusions along the lines of 99% of starting values greater than 1 quadrillion eventually reach a value below 200.&lt;p&gt;Isn&amp;#x27;t this equivalent to saying that &amp;quot;99% of starting values greater than 1 quadrillion eventually reach 1&amp;quot;?&lt;p&gt;I mean, once you reach a value below 200 then you will continue and reach 1. Not only below 200, but below any limit that was experimentally verified (i.e. around 10^20)</text></item></parent_chain><comment><author>MichaelBurge</author><text>If you choose giant starting numbers(like &amp;quot;Tree(3)&amp;quot;), then maybe &amp;quot;quite close&amp;quot; is 10^100.</text></comment>