FACTORINT

This function decomposes an integer into prime factors and their exponents.

For a nonzero integer n, the factorization is represented as:

n = \prod_{k=1}^{m} p_k^{e_k}

where each p_k is prime and each e_k is a positive integer. The output is a 2D table with one row per factor.

Excel Usage

=FACTORINT(n, limit, use_trial, use_rho, use_pm_one, use_ecm, verbose)

n (int, required): Integer to factor.
limit (int, optional, default: null): Optional trial factorization limit.
use_trial (bool, optional, default: true): Whether to use trial division.
use_rho (bool, optional, default: true): Whether to use Pollard rho factor search.
use_pm_one (bool, optional, default: true): Whether to use Pollard p minus 1 method.
use_ecm (bool, optional, default: true): Whether to use elliptic curve method steps.
verbose (bool, optional, default: false): Whether to print verbose factorization progress.

Returns (list[list]): Two-column 2D array of [prime_factor, exponent] rows.

Example 1: Factorization of sixty

Inputs:

n	limit	use_trial	use_rho	use_pm_one	use_ecm	verbose
60		true	true	true	true	false

Excel formula:

=FACTORINT(60, , TRUE, TRUE, TRUE, TRUE, FALSE)

Expected output:

Result
2	2
3	1
5	1

Example 2: Factorization of a prime integer

Inputs:

n	limit	use_trial	use_rho	use_pm_one	use_ecm	verbose
97		true	true	true	true	false

Excel formula:

=FACTORINT(97, , TRUE, TRUE, TRUE, TRUE, FALSE)

Expected output:

Result
97	1

Example 3: Factorization of one thousand two hundred sixty

Inputs:

n	limit	use_trial	use_rho	use_pm_one	use_ecm	verbose
1260		true	true	true	true	false

Excel formula:

=FACTORINT(1260, , TRUE, TRUE, TRUE, TRUE, FALSE)

Expected output:

Result
2	2
3	2
5	1
7	1

Example 4: Factorization including sign for negative input

Inputs:

n	limit	use_trial	use_rho	use_pm_one	use_ecm	verbose
-84		true	true	true	true	false

Excel formula:

=FACTORINT(-84, , TRUE, TRUE, TRUE, TRUE, FALSE)

Expected output:

Result
-1	1
2	2
3	1
7	1

Python Code

from sympy import factorint as sympy_factorint

def factorint(n, limit=None, use_trial=True, use_rho=True, use_pm_one=True, use_ecm=True, verbose=False):
    """
    Compute the prime factorization of an integer with multiplicities.

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

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

    Args:
        n (int): Integer to factor.
        limit (int, optional): Optional trial factorization limit. Default is None.
        use_trial (bool, optional): Whether to use trial division. Default is True.
        use_rho (bool, optional): Whether to use Pollard rho factor search. Default is True.
        use_pm_one (bool, optional): Whether to use Pollard p minus 1 method. Default is True.
        use_ecm (bool, optional): Whether to use elliptic curve method steps. Default is True.
        verbose (bool, optional): Whether to print verbose factorization progress. Default is False.

    Returns:
        list[list]: Two-column 2D array of [prime_factor, exponent] rows.
    """
    try:
        factors = sympy_factorint(
            n,
            limit=limit,
            use_trial=use_trial,
            use_rho=use_rho,
            use_pm1=use_pm_one,
            use_ecm=use_ecm,
            verbose=verbose,
            visual=False,
            multiple=False,
        )

        rows = []
        for factor, exponent in sorted(factors.items(), key=lambda item: int(item[0])):
            rows.append([int(factor), int(exponent)])

        return rows
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

n *

Integer to factor.

limit

Optional trial factorization limit.

use_trial

Whether to use trial division.

use_rho

Whether to use Pollard rho factor search.

use_pm_one

Whether to use Pollard p minus 1 method.

use_ecm

Whether to use elliptic curve method steps.

verbose

Whether to print verbose factorization progress.