https://en.formulasearchengine.com/index.php?action=history&feed=atom&title=2-sided 2026-04-19T17:35:50Z Revision history for this page on the wiki MediaWiki 1.43.0-wmf.28 https://en.formulasearchengine.com/index.php?title=2-sided&diff=13553&oldid=prev 2009-12-16T04:21:01Z

<p>remove Erik9bot category,outdated, tag and general fixes</p> <p><b>New page</b></p><div>{{dablink|You might be looking for [[Legendre polynomials|Legendre&#039;s differential equation]].}}<br /> <br /> In mathematics, &#039;&#039;&#039;Legendre&#039;s equation&#039;&#039;&#039; is the [[Diophantine equation]]<br /> <br /> :&lt;math&gt;ax^2+by^2+cz^2=0.&lt;/math&gt;<br /> <br /> The equation is named for [[Adrien Marie Legendre]] who proved in 1785 that it is solvable in integers &#039;&#039;x&#039;&#039;, &#039;&#039;y&#039;&#039;, &#039;&#039;z&#039;&#039;, not all zero, if and only if<br /> &amp;minus;&#039;&#039;bc&#039;&#039;, &amp;minus;&#039;&#039;ca&#039;&#039; and &amp;minus;&#039;&#039;ab&#039;&#039; are [[quadratic residue]]s modulo &#039;&#039;a&#039;&#039;, &#039;&#039;b&#039;&#039; and &#039;&#039;c&#039;&#039;, respectively, where &#039;&#039;a&#039;&#039;, &#039;&#039;b&#039;&#039;, &#039;&#039;c&#039;&#039; are nonzero, [[Square-free integer|square-free]], [[Pairwise coprime|pairwise relatively prime integers]], not all positive or all negative .<br /> <br /> ==References==<br /> * [[L. E. Dickson]], &#039;&#039;[[History of the Theory of Numbers]]. Vol.II: Diophantine Analysis&#039;&#039;, [[Chelsea Publishing]], 1971, ISBN 0-8284-0086-5. Chap.XIII, p.422.<br /> * J.E. Cremona and D. Rusin, &quot;Efficient solution of rational conics&quot;, [[Mathematics of Computation|Math. Comp.]], &#039;&#039;&#039;72&#039;&#039;&#039; (2003) pp.1417-1441. [http://www.warwick.ac.uk/staff/J.E.Cremona/papers/conics.pdf]<br /> <br /> [[Category:Diophantine equations]]<br /> <br /> {{numtheory-stub}}</div>

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