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numtheory(deprecated)

 legendre
 Legendre symbol

 Calling Sequence legendre(a, p)

Parameters

 a - integer p - prime

Description

 • Important: The numtheory package has been deprecated.  Use the superseding command NumberTheory[LegendreSymbol] instead.
 • The legendre(a, p) function computes the Legendre symbol $L\left(\frac{a}{p}\right)$ of a and p, which is defined to be $1$ if a is a quadratic residue $\mathbf{mod}p$, $-1$ if a is a quadratic non-residue $\mathbf{mod}p$, and $0$ if a is congruent to $0\mathbf{mod}p$. The number a is a quadratic residue mod p if it is not a multiple of p and has a square root $\mathbf{mod}p$, that is, there is an integer $c$ such that ${c}^{2}$ is congruent to $a\mathbf{mod}p$. The number a is a quadratic non-residue mod p if it is not a multiple of p and does not have a square root $\mathbf{mod}p$.
 Note: The legendre routine returns unevaluated if the given algebraic arguments are not of the types specified above.
 • The command with(numtheory,legendre) allows the use of the abbreviated form of this command.

Examples

Important: The numtheory package has been deprecated.  Use the superseding command NumberTheory[LegendreSymbol] instead.

 > $\mathrm{with}\left(\mathrm{numtheory}\right):$
 > $\mathrm{legendre}\left(74,101\right)$
 ${-1}$ (1)
 > $\mathrm{legendre}\left(3,73\right)$
 ${1}$ (2)
 > $\mathrm{legendre}\left(22,11\right)$
 ${0}$ (3)
 > $\mathrm{legendre}\left(5,2\right)$
 ${-1}$ (4)
 > $\mathrm{legendre}\left(-2342,1901\right)$
 ${1}$ (5)
 > $\mathrm{legendre}\left(a,p\right)$
 ${\mathrm{legendre}}{}\left({a}{,}{p}\right)$ (6)

 See Also