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For questions involving the Legendre symbol, $\genfrac{(}{)}{}{}{a}{p}$ for integer $a$ and prime $p$.

In number theory, the Legendre symbol is a multiplicative function with values $1, −1, 0$ that is a quadratic character modulo a prime number $p$: its value on a (nonzero) quadratic residue mod p is 1 and on a non-quadratic residue (non-residue) is $−1$. Its value on zero is $0$.

The Legendre symbol was introduced by Adrien-Marie Legendre in 1798 in the course of his attempts at proving the law of quadratic reciprocity. Generalizations of the symbol include the Jacobi symbol and Dirichlet characters of higher order. The notational convenience of the Legendre symbol inspired introduction of several other "symbols" used in algebraic number theory, such as the Hilbert symbol and the Artin symbol.


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