NTHROOTMOD

This function solves congruences of the form x^n \equiv a \pmod p and returns matching roots in a spillable Excel format.

The equation is:

x^n \equiv a \pmod p

Depending on arithmetic structure, there may be zero, one, or multiple roots. The output is normalized to a single-column 2D array.

Excel Usage

=NTHROOTMOD(a, n, p, all_roots)

a (int, required): Right-hand-side residue value.
n (int, required): Positive exponent.
p (int, required): Positive modulus.
all_roots (bool, optional, default: true): If true, return all roots from the solver.

Returns (list[list]): Single-column 2D array of nth roots modulo p, or a single blank cell if none exist.

Example 1: Documented nth root example returning one root

Inputs:

a	n	p	all_roots
11	4	19	false

Excel formula:

=NTHROOTMOD(11, 4, 19, FALSE)

Expected output:

8

Example 2: Documented nth root example returning all roots

Inputs:

a	n	p	all_roots
11	4	19	true

Excel formula:

=NTHROOTMOD(11, 4, 19, TRUE)

Expected output:

Result
8
11

Example 3: Cubic root congruence example

Inputs:

a	n	p	all_roots
68	3	109	true

Excel formula:

=NTHROOTMOD(68, 3, 109, TRUE)

Expected output:

Result
23
32
54

Example 4: No nth root solution exists

Inputs:

a	n	p	all_roots
2	2	7	true

Excel formula:

=NTHROOTMOD(2, 2, 7, TRUE)

Expected output:

Result
3
4

Python Code

from sympy import nthroot_mod as sympy_nthroot_mod

def nthrootmod(a, n, p, all_roots=True):
    """
    Solve nth-power congruences modulo an integer.

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

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

    Args:
        a (int): Right-hand-side residue value.
        n (int): Positive exponent.
        p (int): Positive modulus.
        all_roots (bool, optional): If true, return all roots from the solver. Default is True.

    Returns:
        list[list]: Single-column 2D array of nth roots modulo p, or a single blank cell if none exist.
    """
    try:
      roots = sympy_nthroot_mod(a, n, p, all_roots=all_roots)
      if roots is None:
        return [[""]]
      if isinstance(roots, int):
        return [[int(roots)]]
      if len(roots) == 0:
        return [[""]]
      return [[int(r)] for r in roots]
    except Exception as e:
      return f"Error: {str(e)}"

Online Calculator

a *

Right-hand-side residue value.

n *

Positive exponent.

p *

Positive modulus.

all_roots

If true, return all roots from the solver.