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

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== Epimericity ==

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 epimericity is also cents (''~~n'' - ''d~~'' )~~/~~cents~~ (~~''d''~~). Then it appears to be true that [[wikipedia: størmer'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 or equal to 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.

If ''n''/''d'' > 1 is a rational number in simplest form, we may define the ''epimericity'' of ''n''/''d'' in terms of cents as cents(''n'' - ''d'')/cents(''d'') - that is, for an example 9/7, n-d is 2 and d is 7, so we end up with roughly 1200/3369 or 0.356. (Note that using other logarithms is possible). Then it appears to be true that [[wikipedia: størmer'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 or equal to 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.

== ''abc'' conjecture ==

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 {{DISPLAYTITLE:''abc'', high quality commas, and epimericity}}
 == Epimericity ==
-If ''n''/''d'' &gt; 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 epimericity is also cents (''n'' - ''d'' )/cents (''d''). Then it appears to be true that [[wikipedia: størmer'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 or equal to 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.
+If ''n''/''d'' &gt; 1 is a rational number in simplest form, we may define the ''epimericity'' of ''n''/''d'' in terms of cents as  cents(''n'' - ''d'')/cents(''d'') - that is, for an example 9/7, n-d is 2 and d is 7, so we end up with roughly 1200/3369 or 0.356. (Note that using other logarithms is possible). Then it appears to be true that [[wikipedia: størmer'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 or equal to 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.
 == ''abc'' conjecture ==