LEGENDRE

This function evaluates the Legendre symbol (a/p) for an integer a and an odd prime p. It classifies whether a is a quadratic residue modulo p.

The value is defined by:

\left(\frac{a}{p}\right)= \begin{cases} 0, & p \mid a \\ 1, & a \text{ is a quadratic residue mod } p \\ -1, & a \text{ is a quadratic nonresidue mod } p \end{cases}

This test is fundamental in modular arithmetic and finite field computations.

Excel Usage

=LEGENDRE(a, p)

a (int, required): Integer whose residue class is tested.
p (int, required): Odd prime modulus.

Returns (int): Legendre symbol value -1, 0, or 1.

Example 1: Legendre symbol zero case

Inputs:

a	p
0	7

Excel formula:

=LEGENDRE(0, 7)

Expected output:

0

Example 2: Legendre symbol residue case

Inputs:

a	p
2	7

Excel formula:

=LEGENDRE(2, 7)

Expected output:

1

Example 3: Legendre symbol nonresidue case

Inputs:

a	p
3	7

Excel formula:

=LEGENDRE(3, 7)

Expected output:

-1

Example 4: Equivalent residue classes produce same value

Inputs:

a	p
10	7

Excel formula:

=LEGENDRE(10, 7)

Expected output:

-1

Python Code

from sympy import legendre_symbol as sympy_legendre_symbol

def legendre(a, p):
    """
    Compute the Legendre symbol modulo an odd prime.

    See: https://docs.sympy.org/latest/modules/ntheory.html#sympy.ntheory.residue_ntheory.legendre_symbol

    This example function is provided as-is without any representation of accuracy.

    Args:
        a (int): Integer whose residue class is tested.
        p (int): Odd prime modulus.

    Returns:
        int: Legendre symbol value -1, 0, or 1.
    """
    try:
        return int(sympy_legendre_symbol(a, p))
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

a *

Integer whose residue class is tested.

p *

Odd prime modulus.