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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>
 
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>
 
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>
* 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;
* 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.
 
== Notes ==
{{reflist}}
 
== See also ==
* [[Fermat's two-square theorem]]
 
[[Category:Additive number theory]]
[[Category:Theorems in number theory]]
 
{{numtheory-stub}}

Latest revision as of 03:25, 24 September 2014

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User friendly CPanel Dashboard.

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I specifically need with any hosting service that I make use of can set up WordPress with quite much a single click or a series of easy clicks. HostGator provides Fantastico De Luxe and QuickInstall choices for quickly installing WordPress and many other scripts to construct your website (i.e. Joomla and others).

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Customer support.

The operators for technical support understand a lot about working in HostGator. If you have a question about personalizing a WordPress theme, HostGator will not aid you (I discovered this to be the case with Bluehost.

1 Objection of HostGator.

I had to call HostGator to validate my account upon opening it. This didn't take long, but it was an extra action. I would have chosen simply to sign up and start without having to call them for verification.

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