152.119.255.250: Couldn't write the citation according to Wikipedia style. But a quick look in the index and you'll find the right page number.

2013-10-07T17:14:08Z

Couldn't write the citation according to Wikipedia style. But a quick look in the index and you'll find the right page number.

← Older revision		Revision as of 19:14, 7 October 2013
Line 1:		Line 1:
	~~Andrew Berryhill is what his wife loves~~ to ~~call him~~ and ~~he totally digs that title~~. ~~My day occupation~~ is ~~an information officer but I~~'~~ve currently utilized for another one~~. ~~North Carolina~~ is ~~where we~~'~~ve been living for many years~~ and ~~will never move~~. ~~I am truly fond~~ of to ~~go to karaoke but I~~'~~ve been taking on new issues recently~~.<br><br>~~my web blog; [http:~~//~~www~~.~~octionx~~.~~sinfauganda~~.co.ug/~~node~~/~~22469 cheap psychic readings~~]		In [[number theory]], '''Tunnell's theorem''' gives a partial resolution to the [[congruent number problem]], and under the [[Birch and Swinnerton-Dyer conjecture]], a full resolution. The congruent number problem asks which [[positive integers]]s can be the area of a right triangle with all three sides rational. Tunnell's theorem relates this to the number of integral solutions to a few fairly simple [[Diophantine equation]]. The theorem is named for [[Jerrold B. Tunnell]], a number theorist at [[Rutgers University]], who proved it in 1983.

			==Theorem==
			For a given square-free integer ''n'', define

			:<math>\begin{matrix}
			A_n & = & \#\{ (x,y,z) \in \mathbb{Z}^3 \| n = 2x^2 + y^2 + 32z^2 \} \\
			B_n & = & \#\{ (x,y,z) \in \mathbb{Z}^3 \| n = 2x^2 + y^2 + 8z^2 \} \quad \\
			C_n & = & \#\{ (x,y,z) \in \mathbb{Z}^3 \| n = 8x^2 + 2y^2 + 64z^2 \} \\
			D_n & = & \#\{ (x,y,z) \in \mathbb{Z}^3 \| n = 8x^2 + 2y^2 + 16z^2 \}.
			\end{matrix}</math>

			Tunnell's theorem states that supposing ''n'' is a congruent number, if ''n'' is odd then 2''A''<sub>''n''</sub> = ''B''<sub>n</sub> and if ''n'' is even then 2''C''<sub>''n''</sub> = ''D''<sub>''n''</sub>. Conversely, if the [[Birch and Swinnerton-Dyer conjecture]] holds true for [[elliptic curve]]s of the form <math>y^2 = x^3 - n^2x</math>, these equalities are sufficient to conclude that ''n'' is a congruent number.

			The importance of Tunnell's theorem is that the criterion it gives is testable by a finite calculation. For instance, for a given ''n'', the numbers ''A''<sub>''n''</sub>,''B''<sub>''n''</sub>,''C''<sub>''n''</sub>,''D''<sub>''n''</sub> can be calculated by exhaustively searching through ''x'',''y'',''z'' in the range <math>-\sqrt{n},\ldots,\sqrt{n}</math>.

			==References==
			* {{cite book
			\| author = Koblitz, Neal
			\| authorlink = Neal Koblitz
			\| title = Introduction to Elliptic Curves and Modular Forms
			\| publisher = Graduate Texts in Mathematics, no. 97, Springer-Verlag
			\| year = 1984
			\| isbn = 0-387-97966-2}}
			* {{cite journal
			\| author = Tunnell, Jerrold B.
			\| authorlink = Jerrold B. Tunnell
			\| title = A classical Diophantine problem and modular forms of weight 3/2
			\| journal = [[Inventiones Mathematicae]]
			\| volume = 72
			\| issue = 2
			\| pages = 323–334
			\| year = 1983
			\| doi = 10.1007/BF01389327}}

			[[Category:Theorems in number theory]]

			{{numtheory-stub}}

en>Helpful Pixie Bot: ISBNs (Build KE)

2012-05-09T02:54:43Z

ISBNs (Build KE)

New page

Andrew Berryhill is what his wife loves to call him and he totally digs that title. My day occupation is an information officer but I've currently utilized for another one. North Carolina is where we've been living for many years and will never move. I am truly fond of to go to karaoke but I've been taking on new issues recently.<br><br>my web blog; [http://www.octionx.sinfauganda.co.ug/node/22469 cheap psychic readings]

DFFITS - Revision history

152.119.255.250: Couldn't write the citation according to Wikipedia style. But a quick look in the index and you'll find the right page number.

en>Helpful Pixie Bot: ISBNs (Build KE)