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| '''Legendre's three-square theorem''' states that any [[natural number]] that is not of the form <math>p = 4^a(8b + 7)</math> for integers ''a'' and ''b'' can be represented as the sum of three integer squares:
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| :<math>p = x^2 + y^2 + z^2\ </math>
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| This theorem was stated by [[Adrien-Marie Legendre]] in 1798.<ref>Conway. Universal Quadratic Forms and the Fifteen Theorem. [http://www.fen.bilkent.edu.tr/~franz/mat/15.pdf]</ref> His proof was incomplete, leaving a gap which was later filled by [[Carl Friedrich Gauss]].<ref>{{cite journal |last1=Dietmann |first1=Rainer |first2=Christian |last2=Elsholtz |title=Sums of two squares and one biquadrate |journal=Funct. Approx. Comment. Math |volume=38 |number=2 |year=2008 |pages=233-234}}</ref>
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| This theorem leads to an easy proof of [[Lagrange's four-square theorem]], which states that all natural numbers can be written as a sum of four squares. Let ''p'' be a natural number, then there are two cases:<ref>{{cite web |author=France Dacar |year=2012 |title=The three squares theorem & enchanted walks |publisher=[[Jozef Stefan Institute]] |accessdate=6 October 2013 |url=http://dis.ijs.si/France/notes/the-three-squares-theorem.pdf}}</ref>
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| * either ''p'' is not of the form <math>4^a(8b + 7)</math>, in which case it is a sum of three squares and thus of four squares <math>p = x^2 + y^2 + z^2 + 0</math> for some ''x'', ''y'', ''z'', by Legendre–Gauss;
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| * or <math>p = 4^a(8b + 7) = (2^a)^2((8b + 6) + 1)</math>, where <math>8b + 6 = 2(4b+3)</math>, which is again a sum of three squares by Legendre–Gauss, so that ''p'' is a sum of four squares.
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| == Notes == | |
| {{reflist}}
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| == See also ==
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| * [[Fermat's two-square theorem]]
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| [[Category:Additive number theory]]
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| [[Category:Theorems in number theory]]
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| {{numtheory-stub}}
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Revision as of 02:59, 26 February 2014
My name's Lance Oneal but everybody calls me Lance. I'm from France. I'm studying at the university (1st year) and I play the Cello for 4 years. Usually I choose songs from my famous films :).
I have two sister. I like Mineral collecting, watching TV (Doctor Who) and Photography.
my site: Hostgator Promo Codes