Abc, high quality commas, and epimericity: Difference between revisions

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=The DoReMi conjecture=

Since not much musical meaning seems to attach to the commas dividing n-d, it makes sense for our purposes to modify the definition of quality. Let doremi(n/d) = log(n)/log((n-d)radical(nd)), where radical(nd) is the product of the primes dividing nd. Then q(n/d) ≤ doremi(n/d), so that the condition that doremi(n/d) &gt; 1+ϵ is stronger than q(n/d) &gt; 1+ϵ, and there will be fewer intervals which qualify. This means that if the list of q(n/d) &gt; 1+ϵ is finite, so is the list of doremi(n/d) &gt; 1+ϵ, and so ABC implies DoReMi but not conversely; DoReMi is a slightly weaker conjecture, but (according to Noam Elkies) still unproven also. Aside from making more musical sense, doremi has the further advantage of being enormously easier to compute if n/d is in some small p-limit, as then the computation of radical(nd) involves only small primes. A comma n/d with doremi(n/d) &gt; 1 we may call a doremi comma.

=Links=  

&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="The DoReMi conjecture"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;The DoReMi conjecture&lt;/h1&gt;

Since not much musical meaning seems to attach to the commas dividing n-d, it makes sense for our purposes to modify the definition of quality. Let doremi(n/d) = log(n)/log((n-d)radical(nd)), where radical(nd) is the product of the primes dividing nd. Then q(n/d) ≤ doremi(n/d), so that the condition that doremi(n/d) &amp;gt; 1+ϵ is stronger than q(n/d) &amp;gt; 1+ϵ, and there will be fewer intervals which qualify. This means that if the list of q(n/d) &amp;gt; 1+ϵ is finite, so is the list of doremi(n/d) &amp;gt; 1+ϵ, and so ABC implies DoReMi but not conversely; DoReMi is a slightly weaker conjecture, but (according to Noam Elkies) still unproven also. Aside from making more musical sense, doremi has the further advantage of being enormously easier to compute if n/d is in some small p-limit, as then the computation of radical(nd) involves only small primes. A comma n/d with doremi(n/d) &amp;gt; 1 we may call a doremi comma.&lt;br /&gt;

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&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Links&lt;/h1&gt;