Abc, high quality commas, and epimericity - Revision history

VectorGraphics at 02:37, 14 April 2025

2025-04-14T02:37:04Z

← Older revision		Revision as of 02:37, 14 April 2025
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	{{DISPLAYTITLE:''abc'', high quality commas, and epimericity}}		{{DISPLAYTITLE:''abc'', high quality commas, and epimericity}}
	== Epimericity ==		== Epimericity ==
	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, but the result is still the same due to the bases cancelling out). 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, but the result is still the same due to the bases cancelling out). 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 a particular finite subset of 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; these are considered "interesting" according to a value judgment whose nature is not known.

	== ''abc'' conjecture ==		== ''abc'' conjecture ==

VectorGraphics at 02:35, 14 April 2025

2025-04-14T02:35:49Z

← Older revision		Revision as of 02:35, 14 April 2025
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	{{DISPLAYTITLE:''abc'', high quality commas, and epimericity}}		{{DISPLAYTITLE:''abc'', high quality commas, and epimericity}}
	== Epimericity ==		== Epimericity ==
	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.		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, but the result is still the same due to the bases cancelling out). 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 ==		== ''abc'' conjecture ==

VectorGraphics at 02:35, 14 April 2025

2025-04-14T02:35:04Z

← Older revision		Revision as of 02:35, 14 April 2025
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	{{DISPLAYTITLE:''abc'', high quality commas, and epimericity}}		{{DISPLAYTITLE:''abc'', high quality commas, and epimericity}}
	== 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 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 ==		== ''abc'' conjecture ==

Sintel: -legacy, todo intro, inaccessible

2025-03-29T17:55:26Z

-legacy, todo intro, inaccessible

← Older revision		Revision as of 17:55, 29 March 2025
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			{{inacc}}
			{{todo\|intro\|inline=1}}

	{{DISPLAYTITLE:''abc'', high quality commas, and epimericity}}		{{DISPLAYTITLE:''abc'', high quality commas, and epimericity}}
	== Epimericity ==		== Epimericity ==

Sintel: -legacy

2025-03-29T17:53:41Z

-legacy

← Older revision		Revision as of 17:53, 29 March 2025
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	{{DISPLAYTITLE:''abc'', high quality commas, and epimericity}}		{{DISPLAYTITLE:''abc'', high quality commas, and epimericity}}
	~~{{Legacy}}~~
	== 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.

Lériendil at 16:44, 17 February 2025

2025-02-17T16:44:38Z

← Older revision		Revision as of 16:44, 17 February 2025
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	{{DISPLAYTITLE:''abc'', high quality commas, and epimericity}}		{{DISPLAYTITLE:''abc'', high quality commas, and epimericity}}
	{{Legacy~~\|Abc,_high_quality_commas,_and_epimericity~~}}		{{Legacy}}
	== 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.

Lériendil: deploying new cat

2025-02-17T16:20:53Z

deploying new cat

← Older revision		Revision as of 16:20, 17 February 2025
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	{{DISPLAYTITLE:''abc'', high quality commas, and epimericity}}		{{DISPLAYTITLE:''abc'', high quality commas, and epimericity}}
			{{Legacy\|Abc,_high_quality_commas,_and_epimericity}}
	== 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.

Lériendil: Changed protection level for "Abc, high quality commas, and epimericity" ([Edit=Allow only administrators] (expires 17:04, 17 February 2025 (UTC)) [Move=Allow only administrators] (indefinite))

2025-02-17T16:04:21Z

Changed protection level for "Abc, high quality commas, and epimericity" ([Edit=Allow only administrators] (expires 17:04, 17 February 2025 (UTC)) [Move=Allow only administrators] (indefinite))

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FloraC: Recategorize

2024-12-18T10:40:16Z

Recategorize

← Older revision		Revision as of 10:40, 18 December 2024
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	[[Category:Math]]		[[Category:Math]]

	{{Todo\| ~~add introduction~~ }}		{{Todo\| intro }}

Mike Battaglia: Protected "Abc, high quality commas, and epimericity" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

2021-11-15T03:25:23Z

Protected "Abc, high quality commas, and epimericity" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

← Older revision	Revision as of 03:25, 15 November 2021
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