British Association screw threads - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10514)

2014-12-12T19:33:38Z

WP:CHECKWIKI error fixes using AWB (10514)

← Older revision		Revision as of 21:33, 12 December 2014
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en>Rhododendrites: Reverted 2 good faith edits by 70.26.15.69 using STiki -- comments on the talk page please

2014-02-23T07:52:26Z

Reverted 2 good faith edits by 70.26.15.69 using STiki -- comments on the talk page please

← Older revision		Revision as of 09:52, 23 February 2014
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	~~In [[mathematics]], in the field of [[number theory]], the '''Ramanujan–Nagell equation'''~~ is ~~a particular~~ [~~[exponential Diophantine equation]]~~.		Emilia Shryock is home std test my name but you can [http://stdfreeclinics.com/SYPHILIS.html contact] me something you like. To gather badges is what her std home test family members and her appreciate. North Dakota is exactly where me and [http://www.dynast.co.kr/xe/?document_srl=150618 at home std test] my spouse live. Since she was eighteen she's been working as a meter reader but [http://www.biogids.nl/biobank/2014-06-13/how-can-1-especially-get-around-todays-diseases at home std testing] she's always needed her personal business.<br><br>Look at my web site :: home [http://Funnie.st/365631/celebs-who-had-herpes/ std test] kit ([http://test.ithink-now.org/content/think-f83d96d08fe71e25bf80a6cb59048a18 Read the Full Piece of writing])

	~~==Equation and solution==~~
	~~The equation is~~
	~~:<math>2^n-7=x^2 \,<~~/~~math>~~
	~~and solutions in natural numbers ''n'' and ''x'' exist just when ''n'' = 3, 4, 5, 7 and 15~~.

	This was conjectured in 1913 by Indian mathematician [[Srinivasa Ramanujan]], proposed independently in 1943 by the Norwegian mathematician [[Wilhelm Ljunggren]], and [[mathematical proof\|proved]] in 1948 by the Norwegian mathematician [[Trygve Nagell]]. ~~The values on ''n'' correspond to the values of ''x'' as:-~~

	~~:''x'' = 1, 3, 5, 11~~ and ~~181.<ref>Saradha & Srinivasan (2008) p~~.~~208</ref>~~

	~~== Triangular Mersenne numbers ==~~
	~~The problem of finding all numbers of the form 2<sup>''b''</sup> − 1 ([[Mersenne number]]s) which are [[triangular number\|triangular]]~~ is ~~equivalent:~~

	~~:<math>2^b-1 = \frac{y(y+1)}{2}</math>~~
	~~:<math>\Leftrightarrow 8(2^b-1) = 4y(y+1)</math>~~
	~~:<math>\Leftrightarrow 2^{b+3}-8 = 4y^2+4y</math>~~
	~~:<math>\Leftrightarrow 2^{b+3}-7 = 4y^2+4y+1</math>~~
	~~:<math>\Leftrightarrow 2^{b+3}-7 = (2y+1)^2</math>~~

	~~The values of ''b'' are just those of ''n'' − 3,~~ and ~~the corresponding triangular Mersenne numbers (also known as '''Ramanujan–Nagell numbers''') are~~:

	~~:<math>\frac{y(y+1)}{2} = \frac{(x-1)(x+1)}{8}<~~/~~math>~~

	~~for ''x'' = 1, 3, 5, 11 and 181, giving 0, 1, 3, 15, 4095 and no more {{OEIS\|id=A076046}}.~~

	~~==Equations of Ramanujan–Nagell type==~~
	~~An equation of the form~~

	~~:<math> x^2 + D = A B^n <~~/~~math>~~

	~~for fixed ''D'', ''A'' , ''B'' and variable ''x'', ''n'' is said to be of ''Ramanujan–Nagell type''~~. ~~A result of [[Carl Ludwig Siegel\|Siegel]] implies that the number of solutions in each case is finite~~.~~<ref>Saradha & Srinivasan (2008) p~~.~~207<~~/~~ref> The equation with ''A''=1, ''B''~~=~~2 has~~ at ~~most two solutions except in the case~~ '~~'D''=7 already mentioned.~~ ~~There are infinitely many values of ''D'' for which there are two solutions, including <math>D = 2^m - 1</math>.<ref>Saradha & Srinivasan (2008) p.208</ref>~~

	~~==Equations of Lebesgue–Nagell type==~~
	~~An equation of the form~~

	:~~<math> x^2 + D = A y^n <~~/~~math>~~

	for fixed ''D'', ''A'' and variable ''x'', ''y'', ''n'' is said to be of ''Lebesgue–Nagell type''. Results of Shorey and [[Robert Tijdeman\|Tijdeman]] imply that the number of solutions in each case is finite.<ref>Saradha & Srinivasan (2008) p.211</~~ref>~~

	~~==See also==~~
	* [[Scientific equations named after people]]

	~~==References==~~
	~~{{reflist}}~~
	* {{cite journal \| author=S. Ramanujan \| authorlink=Srinivasa Ramanujan \| title=Question 464 \| journal=J. Indian Math. Soc. \| volume=5 \| year=1913 \| pages=130 }}
	* {{cite journal \| author=W. Ljunggren \| authorlink=WIlhelm Ljunggren \| title=Oppgave nr 2 \| journal=Norsk Mat. Tidsskr. \| volume=25 \| year=1943 \| pages=29 }}
	* {{cite journal \| author=T. Nagell \| title=Løsning till oppgave nr 2 \| journal=Norsk Mat. ~~Tidsskr~~. ~~\| volume=30 \| year=1948 \| pages=62–64 }}~~
	* {{cite journal \| author=T. Nagell \| title=The Diophantine equation ''x''<sup>2</sup>+7=2<sup>''n''</~~sup> \| journal=Ark. Mat. \| volume=30 \| year=1961 \| pages=185–187 \| doi=10.1007~~/~~BF02592006 }}~~
	* {{cite book \| last1=Shorey \| first1=T.N. \| last2=Tijdeman \| first2=R. \| author2-~~link=Robert Tijdeman \| title=Exponential Diophantine equations \| series=Cambridge Tracts in Mathematics \| volume=87 \| publisher=[[Cambridge University Press]] \| year=1986 \| isbn=0~~-~~521~~-~~26826~~-~~5 \| zbl=0606.10011 \| pages=137–138 }}~~
	* {{cite book \| editor-~~first=N. \| editor~~-~~last=Saradha \| title=Diophantine Equations \| publisher=Narosa \| year=2008 \| isbn=978~~-81-~~7319~~-~~898-4 \| first1=N~~. ~~\| last1=Saradha \| first2=Anitha \| last2=Srinivasan \| chapter=Generalized Lebesgue–Ramanujan–Nagell equations \| pages=207–223 }}~~

	~~==External links==~~
	* {{cite web ~~\| url=~~http://~~mathworld~~.~~wolfram.com~~/~~RamanujansSquareEquation.html \| publisher=Wolfram MathWorld \| title=Values of X corresponding to N in the Ramanujan–Nagell Equation \| accessdate=2012~~-05-~~08 }}~~
	* [http://~~mathforum~~.org/~~kb/message.jspa?messageID=419063&tstart=0 Can ''N''<sup>2<~~/~~sup> + ''N'' + 2 Be A Power Of 2?], Math Forum discussion~~

	~~{{DEFAULTSORT:Ramanujan~~-~~Nagell equation}}~~
	~~[[Category:Diophantine equations]]~~
	~~[[Category:Srinivasa Ramanujan]~~]

92.40.78.0 at 01:12, 1 December 2013

2013-12-01T01:12:03Z

← Older revision		Revision as of 03:12, 1 December 2013
Line 1:		Line 1:
	~~Oscar~~ is ~~how he~~'~~s called~~ and ~~he completely enjoys this name~~. ~~North Dakota is her beginning location but she will have~~ to ~~move one day or another~~. ~~He utilized to be unemployed but now he is a meter reader~~. ~~Body building~~ is 1 of the ~~issues I adore most~~.<br><br>~~Feel free~~ to ~~surf to my webpage~~ ... ~~[http~~://~~reinodoprazer~~.~~com~~.~~br/index~~.~~php?do~~=/~~blog~~/~~326~~/~~tips~~-~~about~~-~~how~~-to-~~overcome~~-~~candidiasis~~-~~easily/ home std test kit~~]		In [[mathematics]], in the field of [[number theory]], the '''Ramanujan–Nagell equation''' is a particular [[exponential Diophantine equation]].

			==Equation and solution==
			The equation is
			:<math>2^n-7=x^2 \,</math>
			and solutions in natural numbers ''n'' and ''x'' exist just when ''n'' = 3, 4, 5, 7 and 15.

			This was conjectured in 1913 by Indian mathematician [[Srinivasa Ramanujan]], proposed independently in 1943 by the Norwegian mathematician [[Wilhelm Ljunggren]], and [[mathematical proof\|proved]] in 1948 by the Norwegian mathematician [[Trygve Nagell]]. The values on ''n'' correspond to the values of ''x'' as:-

			:''x'' = 1, 3, 5, 11 and 181.<ref>Saradha & Srinivasan (2008) p.208</ref>

			== Triangular Mersenne numbers ==
			The problem of finding all numbers of the form 2<sup>''b''</sup> − 1 ([[Mersenne number]]s) which are [[triangular number\|triangular]] is equivalent:

			:<math>2^b-1 = \frac{y(y+1)}{2}</math>
			:<math>\Leftrightarrow 8(2^b-1) = 4y(y+1)</math>
			:<math>\Leftrightarrow 2^{b+3}-8 = 4y^2+4y</math>
			:<math>\Leftrightarrow 2^{b+3}-7 = 4y^2+4y+1</math>
			:<math>\Leftrightarrow 2^{b+3}-7 = (2y+1)^2</math>

			The values of ''b'' are just those of ''n'' − 3, and the corresponding triangular Mersenne numbers (also known as '''Ramanujan–Nagell numbers''') are:

			:<math>\frac{y(y+1)}{2} = \frac{(x-1)(x+1)}{8}</math>

			for ''x'' = 1, 3, 5, 11 and 181, giving 0, 1, 3, 15, 4095 and no more {{OEIS\|id=A076046}}.

			==Equations of Ramanujan–Nagell type==
			An equation of the form

			:<math> x^2 + D = A B^n </math>

			for fixed ''D'', ''A'' , ''B'' and variable ''x'', ''n'' is said to be of ''Ramanujan–Nagell type''. A result of [[Carl Ludwig Siegel\|Siegel]] implies that the number of solutions in each case is finite.<ref>Saradha & Srinivasan (2008) p.207</ref> The equation with ''A''=1, ''B''=2 has at most two solutions except in the case ''D''=7 already mentioned. There are infinitely many values of ''D'' for which there are two solutions, including <math>D = 2^m - 1</math>.<ref>Saradha & Srinivasan (2008) p.208</ref>

			==Equations of Lebesgue–Nagell type==
			An equation of the form

			:<math> x^2 + D = A y^n </math>

			for fixed ''D'', ''A'' and variable ''x'', ''y'', ''n'' is said to be of ''Lebesgue–Nagell type''. Results of Shorey and [[Robert Tijdeman\|Tijdeman]] imply that the number of solutions in each case is finite.<ref>Saradha & Srinivasan (2008) p.211</ref>

			==See also==
			* [[Scientific equations named after people]]

			==References==
			{{reflist}}
			* {{cite journal \| author=S. Ramanujan \| authorlink=Srinivasa Ramanujan \| title=Question 464 \| journal=J. Indian Math. Soc. \| volume=5 \| year=1913 \| pages=130 }}
			* {{cite journal \| author=W. Ljunggren \| authorlink=WIlhelm Ljunggren \| title=Oppgave nr 2 \| journal=Norsk Mat. Tidsskr. \| volume=25 \| year=1943 \| pages=29 }}
			* {{cite journal \| author=T. Nagell \| title=Løsning till oppgave nr 2 \| journal=Norsk Mat. Tidsskr. \| volume=30 \| year=1948 \| pages=62–64 }}
			* {{cite journal \| author=T. Nagell \| title=The Diophantine equation ''x''<sup>2</sup>+7=2<sup>''n''</sup> \| journal=Ark. Mat. \| volume=30 \| year=1961 \| pages=185–187 \| doi=10.1007/BF02592006 }}
			* {{cite book \| last1=Shorey \| first1=T.N. \| last2=Tijdeman \| first2=R. \| author2-link=Robert Tijdeman \| title=Exponential Diophantine equations \| series=Cambridge Tracts in Mathematics \| volume=87 \| publisher=[[Cambridge University Press]] \| year=1986 \| isbn=0-521-26826-5 \| zbl=0606.10011 \| pages=137–138 }}
			* {{cite book \| editor-first=N. \| editor-last=Saradha \| title=Diophantine Equations \| publisher=Narosa \| year=2008 \| isbn=978-81-7319-898-4 \| first1=N. \| last1=Saradha \| first2=Anitha \| last2=Srinivasan \| chapter=Generalized Lebesgue–Ramanujan–Nagell equations \| pages=207–223 }}

			==External links==
			* {{cite web \| url=http://mathworld.wolfram.com/RamanujansSquareEquation.html \| publisher=Wolfram MathWorld \| title=Values of X corresponding to N in the Ramanujan–Nagell Equation \| accessdate=2012-05-08 }}
			* [http://mathforum.org/kb/message.jspa?messageID=419063&tstart=0 Can ''N''<sup>2</sup> + ''N'' + 2 Be A Power Of 2?], Math Forum discussion

			{{DEFAULTSORT:Ramanujan-Nagell equation}}
			[[Category:Diophantine equations]]
			[[Category:Srinivasa Ramanujan]]

en>Hyperwie at 00:19, 19 June 2012

2012-06-19T00:19:12Z

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