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Etymology

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Named after French mathematician Adrien-Marie Legendre (1752–1833), who introduced the symbol in 1798 in his work Essai sur la Théorie des Nombres ("Essay on the Theory of Numbers").

Noun

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Legendre symbol (plural Legendre symbols)

  1. (number theory) A mathematical function of an integer and a prime number, written  , which indicates whether a is a quadratic residue modulo p.
    • 1994, James K. Strayer, Elementary Number Theory, Waveland Press, 2002, Reissue, page 109,
      Our only method at present for the computation of Legendre symbols requires a possible consideration of   congruences (unless, of course, we are fortunate enough to encounter the desired quadratic residue along the way).
    • 2006, Neville Robbins, Beginning Number Theory, 2nd edition, Jones & Bartlett, page 195:
      The Jacobi symbol, which generalizes the Legendre symbol, sheds some additional light on how to determine whether (7.29) has solutions when m has two or more distinct prime factors.
    • 2013, Song Y. Yan, Number Theory for Computing, Springer, page 149:
      Jacobi symbols can be used to facilitate the calculation of Legendre symbols. In fact, Legendre symbols can be eventually calculated by Jacobi symbols [17]. That is, the Legendre symbol can be calculated as if it were a Jacobi symbol. For example, consider the Legendre symbol   where 335 = 5·67 is not a prime (of course, 2999 is a prime, otherwise, it is not a Legendre symbol).

Usage notes

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The symbol takes the values:

 

It is generalised to composite numbers by the Jacobi symbol, which is identical in form and range of values and is defined as a product of Legendre symbols.

See also

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