Abc, high quality commas, and epimericity
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2012-09-07 11:44:28 UTC.
- The original revision id was 362842988.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
If n/d > 1 is a rational number with positive integers n and d relatively prime, we may define the //epimericity// of n/d as log(n-d)/log(d). Which logarithm we use is irrelevant; we can if we like use cents and so the peimericity is also cents(n-d)/cents(d). Then it appears to be true that [[http://en.wikipedia.org/wiki/St%C3%B8rmer's_theorem|Størmer's theorem]] generalizes to a claim that for any prime p, only finitely many rational numbers in the p-limit exist with epimericity less than any constant c less than one. Hence "interesting" commas in any p-limit can be defined as those below a given epimericity, such as the 7-limit commas under 0.5 in epimericity, or the 11-limit commas under 0.3. This conjecture is related to the [[http://en.wikipedia.org/wiki/Abc_conjecture|abc conjecture]], and a related claim is in fact precisely the abc conjecture, which defines what we may call a //high quality comma//. Define the //radical// rad(n/d) of n/d as the product of all the primes dividing n, d, and n-d; so that rad(128/125) = 2*3*5 = 30. Then define the //quality// q(n/d) of n/d as log(n-d)/log(rad(n/d)). Then the abc conjecture, a very powerful conjecture, says that for any ϵ>0 there are only finitely many commas such that q(n/d) > 1+ϵ. Any comma with q(n/d)>1 we may call "high quality".
Original HTML content:
<html><head><title>ABC, High Quality Commas, and Epimericity</title></head><body>If n/d > 1 is a rational number with positive integers n and d relatively prime, we may define the <em>epimericity</em> of n/d as log(n-d)/log(d). Which logarithm we use is irrelevant; we can if we like use cents and so the peimericity is also cents(n-d)/cents(d). Then it appears to be true that <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/St%C3%B8rmer's_theorem" rel="nofollow">Størmer's theorem</a> generalizes to a claim that for any prime p, only finitely many rational numbers in the p-limit exist with epimericity less than any constant c less than one. Hence "interesting" commas in any p-limit can be defined as those below a given epimericity, such as the 7-limit commas under 0.5 in epimericity, or the 11-limit commas under 0.3.<br /> <br /> This conjecture is related to the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Abc_conjecture" rel="nofollow">abc conjecture</a>, and a related claim is in fact precisely the abc conjecture, which defines what we may call a <em>high quality comma</em>. Define the <em>radical</em> rad(n/d) of n/d as the product of all the primes dividing n, d, and n-d; so that rad(128/125) = 2*3*5 = 30. Then define the <em>quality</em> q(n/d) of n/d as log(n-d)/log(rad(n/d)). Then the abc conjecture, a very powerful conjecture, says that for any ϵ>0 there are only finitely many commas such that q(n/d) > 1+ϵ. Any comma with q(n/d)>1 we may call "high quality".</body></html>