Legendre's three-square theorem: Difference between revisions
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'''Legendre's three-square theorem''' states that any [[natural number]] that is not of the form <math>4^a(8b + 7)</math> for integers a and b can be represented as the sum of three integer squares |
'''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> |
:<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]]. |
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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== Notes == |
== Notes == |
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[[Category:Additive number theory]] |
[[Category:Additive number theory]] |
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[[Category:Theorems in number theory]] |
[[Category:Theorems in number theory]] |
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Revision as of 15:15, 6 October 2013
Legendre's three-square theorem states that any natural number that is not of the form for integers a and b can be represented as the sum of three integer squares:
This theorem was stated by Adrien-Marie Legendre in 1798.[1] His proof was incomplete, leaving a gap which was later filled by Carl Friedrich Gauss.[2]
Notes
See also
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