66.177.64.39: Corrects ISBN

2013-07-15T19:30:39Z

Corrects ISBN

← Older revision		Revision as of 21:30, 15 July 2013
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	~~Friends contact him Royal Cummins~~. ~~His day job is a financial officer but he plans on changing~~ it. ~~Some time~~ in ~~the past I selected~~ to ~~reside~~ in ~~Arizona but I need to transfer~~ for ~~my family members~~. ~~Bottle tops gathering~~ is the ~~only pastime his wife doesn~~'~~t approve~~ of.<br><br>~~Also visit my homepage~~ ... [http://~~Beta~~.~~Westernstatesfireequipment~~.~~com~~/~~ActivityFeed~~/~~MyProfile~~/~~tabid~~/93/~~userId~~/~~1095~~/~~Default~~.~~aspx Beta~~.~~Westernstatesfireequipment~~.~~com~~]		The '''15 theorem''' of [[John H. Conway]] and W. A. Schneeberger ('''Conway–Schneeberger Fifteen Theorem'''), proved in 1993, states that if an integral [[quadratic form]] with [[integer matrix]] represents all [[positive integer]]s up to [[15 (number)\|15]], then it represents all positive integers. (All quadratic forms in this article are implicitly assumed to be positive definite.) The proof was complicated, and was never published. [[Manjul Bhargava]] found a much simpler proof which was published in 2000.

			This result is the best possible, as there are such forms as, for example,

			:<math>w^2 + 2x^2 + 5y^2 + 5z^2,\ </math>

			that represent all positive integers other than 15.

			A quadratic form representing all positive integers is sometimes called '''universal'''. For example,

			:<math>w^2 + x^2 + y^2 + z^2\ </math>

			is universal because every positive integer can be written as a sum of 4 squares, by [[Lagrange's four-square theorem]]. By the 15 theorem, to verify this it is sufficient to check that every positive integer up to 15 is a sum of 4 squares. (This does not give an alternative proof of Lagrange's theorem, because Lagrange's theorem is used in the proof of the 15 theorem.)

			A more precise version of the 15 theorem says that if an integral quadratic form with integral matrix represents the numbers 1, 2, 3, 5, 6, 7, 10, 14, 15 {{OEIS\|id=A030050}} then it represents all positive integers. Moreover, for each of these 9 numbers, there is such a quadratic form representing all positive integers except for this number.

			There are two different notions of integrality for integral quadratic forms: it can be called integral if its associated symmetric matrix is integral ("integral matrix"), or if all its coefficients are integral ("integer valued"). For example,
			''x''<sup>2</sup> + ''xy'' + ''y''<sup>2</sup> is integer valued but does not have integral matrix.

			In 2005 [[Manjul Bhargava]] and Jonathan P. Hanke announced a proof of Conway's conjecture that a similar [[theorem]] holds for integer valued integral quadratic forms, with the constant 15 replaced by [[290 (number)\|290]]. The proof is to appear in ''Inventiones Mathematicae''.<ref>Bhargava, M., & Hanke, J. Universal quadratic forms and the 290-theorem. ''Invent. Math.'', to appear.</ref>

			A more precise version states that if an integer valued integral quadratic form represents all the numbers 1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290 {{OEIS\|id=A030051}} then it represents all positive integers, and for each of these 29 numbers there is such a quadratic form representing all positive integers with the exception of this one number.

			Bhargava has found analogous criteria for an integral quadratic form to represent all primes (the set {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73} {{OEIS\|id=A154363}}) and for such a quadratic form to represent all positive odd integers (the set {1, 3, 5, 7, 11, 15, 33} {{OEIS\|id=A116582}}).

			==References==
			* {{cite book \| last=Bhargava \| first=Manjul \| authorlink=Manjul Bhargava \| url=http://www.fen.bilkent.edu.tr/~franz/mat/15.pdf \| chapter=On the Conway–Schneeberger fifteen theorem \| title=Quadratic forms and their applications (Dublin, 1999) \| pages=27–37 \| series=Contemp. Math. \| volume=272 \| publisher=Amer. Math. Soc. \| location=Providence, RI \| year=2000 \| isbn=0-8218-2779-0 \| zbl=0987.11027 }}
			* {{cite book \| authorlink=John Horton Conway \| last=Conway \| first=J.H. \| url=http://www.fen.bilkent.edu.tr/~franz/mat/15.pdf \| chapter=Universal quadratic forms and the fifteen theorem \| title=Quadratic forms and their applications (Dublin, 1999) \| pages=23–26 \| series=Contemp. Math. \| volume=272 \| publisher=Amer. Math. Soc. \| location=Providence, RI \| year=2000 \| isbn=0-8218-2779-0 \| zbl=0987.11026 }}
			{{reflist}}

			[[Category:Additive number theory]]
			[[Category:Theorems in number theory]]
			[[Category:Quadratic forms]]

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2012-05-07T00:46:10Z

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Friends contact him Royal Cummins. His day job is a financial officer but he plans on changing it. Some time in the past I selected to reside in Arizona but I need to transfer for my family members. Bottle tops gathering is the only pastime his wife doesn't approve of.<br><br>Also visit my homepage ... [http://Beta.Westernstatesfireequipment.com/ActivityFeed/MyProfile/tabid/93/userId/1095/Default.aspx Beta.Westernstatesfireequipment.com]

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66.177.64.39: Corrects ISBN

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