QUADRES

This function returns all quadratic residues modulo p, i.e., all values attainable as squares in modular arithmetic.

The residue set is:

R_p = \{x^2 \bmod p : x \in \{0,1,\dots,p-1\}\}

The values are returned as a sorted single-column 2D array for Excel range output.

Excel Usage

=QUADRES(p)

p (int, required): Positive modulus.

Returns (list[list]): Single-column 2D array of quadratic residues modulo p.

Example 1: Quadratic residues modulo seven

Inputs:

p
7

Excel formula:

=QUADRES(7)

Expected output:

Result
0
1
2
4

Example 2: Quadratic residues modulo eleven

Inputs:

p
11

Excel formula:

=QUADRES(11)

Expected output:

Result
0
1
3
4
5
9

Example 3: Quadratic residues modulo nine

Inputs:

p
9

Excel formula:

=QUADRES(9)

Expected output:

Result
0
1
4
7

Example 4: Quadratic residues modulo thirteen

Inputs:

p
13

Excel formula:

=QUADRES(13)

Expected output:

Result
0
1
3
4
9
10
12

Python Code

from sympy import quadratic_residues as sympy_quadratic_residues

def quadres(p):
    """
    List all quadratic residues modulo p.

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

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

    Args:
        p (int): Positive modulus.

    Returns:
        list[list]: Single-column 2D array of quadratic residues modulo p.
    """
    try:
        residues = sympy_quadratic_residues(p)
        return [[int(v)] for v in residues]
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

p *

Positive modulus.