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Legendre's equation

Special diophantine equation involving squares From Wikipedia, the free encyclopedia

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In mathematics, Legendre's equation is a Diophantine equation of the form:

The equation is named for Adrien-Marie Legendre who proved it in 1785 that it is solvable in integers x, y, z, not all zero, if and only if bc, ca and ab are quadratic residues modulo a, b and c, respectively, where a, b, c are nonzero, square-free, pairwise relatively prime integers and also not all positive or all negative.

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References

  • L. E. Dickson, History of the Theory of Numbers. Vol.II: Diophantine Analysis, Chelsea Publishing, 1971, ISBN 0-8284-0086-5. Chap.XIII, p. 422.
  • J.E. Cremona and D. Rusin, "Efficient solution of rational conics", Math. Comp., 72 (2003) pp. 1417-1441.


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