Abc, high quality commas, and epimericity: Difference between revisions
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== Epimericity == | == 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 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. | ||
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[[Category:Comma]] | [[Category:Comma]] | ||
[[Category:Math]] | [[Category:Math]] | ||
{{Todo| add introduction }} | |||