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FactorInteger
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FactorInteger

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gives a list of the prime factors of the integer n, together with their exponents.

✖

does partial factorization, pulling out at most k distinct factors.

Details and Options

  • FactorInteger is also known as prime factorization.
  • For a positive number n=p1k1⋯ pmkm with pi primes, FactorInteger[n] gives a list {{p1,k1},…,{pm,km}}.
  • For negative numbers, the unit {-1,1} is included in the list of factors.
  • FactorInteger also works on rational numbers. The prime factors of the denominator are given with negative exponents.
  • FactorInteger[n,GaussianIntegers->True] factors over Gaussian integers.
  • FactorInteger[m+I n] automatically works over Gaussian integers.
  • When necessary, a unit of the form {-1,1}, {I,1} or {-I,1} is included in the list of factors.
  • The last element in the list FactorInteger[n,k] gives what is left after the partial factorization.
  • FactorInteger[n,Automatic] pulls out only factors that are easy to find.
  • FactorInteger uses PrimeQ to determine whether factors are prime.

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Find prime factors of and their exponents:

In[2]:=2
✖
https://wolfram.com/xid/0enx9tara-f4g
Out[2]=2

Plot the number of distinct prime factors of the first 100 numbers:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-6aoqa
Out[1]=1

Scope  (6)Survey of the scope of standard use cases

FactorInteger works over integers:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-h7skky
Out[1]=1

Rational numbers:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-gvzk29
Out[1]=1

Gaussian integers:

In[1]:=1
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https://wolfram.com/xid/0enx9tara-bv9dzr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0enx9tara-bp03jc
Out[2]=2

Find a partial factorization:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-e5bo1e
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0enx9tara-5yquql
Out[2]=2

Compute for large integers:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-dco2wk
Out[1]=1

FactorInteger threads over lists:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-0khk1r
Out[1]=1

Options  (1)Common values & functionality for each option

GaussianIntegers  (1)

Factor a number over integers:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-10e1vr
Out[1]=1

Gaussian integers:

In[2]:=2
✖
https://wolfram.com/xid/0enx9tara-s7a4h8
Out[2]=2

Applications  (12)Sample problems that can be solved with this function

Basic Applications  (5)

Every positive integer can be represented as a product of prime factors:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-lgbdq
Out[1]=1

A unit factor:

In[2]:=2
✖
https://wolfram.com/xid/0enx9tara-0oy1zc
Out[2]=2

Plot the number of distinct prime factors of numbers up to :

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-gdv
Out[1]=1

Compare with the number of distinct prime factors over the Gaussian integers:

In[2]:=2
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https://wolfram.com/xid/0enx9tara-kcb
Out[2]=2

Display as an explicit product of factors:

In[1]:=1
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https://wolfram.com/xid/0enx9tara-v9i
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0enx9tara-bw2
Out[2]=2

Use FactorInteger to test for prime powers:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-efcup7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0enx9tara-uc5qr5
Out[2]=2

Use FactorInteger to find all prime divisors of a number:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-pefmwt
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0enx9tara-ona38t
Out[2]=2

Number Theory  (7)

Use FactorInteger to compute the number of divisors of the number:

In[3]:=3
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https://wolfram.com/xid/0enx9tara-dny2q5
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0enx9tara-jdh9sy
Out[4]=4

Use FactorInteger to recognize powerful numbers, numbers whose prime factors are all repeated:

In[1]:=1
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https://wolfram.com/xid/0enx9tara-3x7iu
In[2]:=2
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https://wolfram.com/xid/0enx9tara-bps1v3
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0enx9tara-g4cjq
Out[3]=3

Find factorizations of numbers of the form :

In[1]:=1
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https://wolfram.com/xid/0enx9tara-c4k
Out[1]=1

Find all natural numbers up to 100 that are primes or prime powers:

In[1]:=1
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https://wolfram.com/xid/0enx9tara-vdr
Out[1]=1

The highest power of a prime in numbers up to 100:

In[1]:=1
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https://wolfram.com/xid/0enx9tara-re0
Out[1]=1

Find primes that appear in prime factorization of only to the first power:

In[1]:=1
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https://wolfram.com/xid/0enx9tara-wpc
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0enx9tara-dd3
Out[2]=2

Use FactorInteger to compute the square-free part of a number:

In[1]:=1
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https://wolfram.com/xid/0enx9tara-kzeujz
In[2]:=2
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https://wolfram.com/xid/0enx9tara-52qujd
Out[2]=2

Visualize the distribution:

In[3]:=3
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https://wolfram.com/xid/0enx9tara-76rbjv
Out[3]=3

Properties & Relations  (9)Properties of the function, and connections to other functions

The prime factorization of a prime number is itself:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-konju4
Out[1]=1
In[3]:=3
✖
https://wolfram.com/xid/0enx9tara-dab0xs
Out[3]=3

Prime powers:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-hbn4a7
Out[1]=1
In[15]:=15
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https://wolfram.com/xid/0enx9tara-bog1xe
Out[15]=15

Composite numbers have at least two prime factors including multiplicities:

In[1]:=1
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https://wolfram.com/xid/0enx9tara-z74y0b
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0enx9tara-krrkpj
Out[2]=2

Compute the original number from a factorization:

In[1]:=1
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https://wolfram.com/xid/0enx9tara-qyepwb
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0enx9tara-ka8wbr
Out[2]=2

Exponents in the prime factorization of a square-free number are all :

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-c5x9z
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0enx9tara-m1yhtl
Out[2]=2

Divisors gives the list of divisors including prime divisors:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-4db6ss
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0enx9tara-je44me
Out[2]=2

PrimeNu gives the number of distinct prime factors:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-9fi76v
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0enx9tara-l073yk
Out[2]=2

PrimeOmega gives the number of prime factors counting multiplicities:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-dosvv2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0enx9tara-2298n
Out[2]=2

Coprime numbers have no prime factors in common:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-mtarak
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0enx9tara-spxt2r
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0enx9tara-vok8bj
Out[3]=3

If the prime factorization of n is given by , then the number of divisors of n is :

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-0hqi2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0enx9tara-6qfue
Out[2]=2

Possible Issues  (2)Common pitfalls and unexpected behavior

Timings can increase rapidly and unpredictably with the size of the input:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-le1
Out[1]=1

FactorInteger at zero:

In[1]:=1
✖
https://wolfram.com/xid/0enx9tara-hkd5ly
Out[1]=1
Wolfram Research (1988), FactorInteger, Wolfram Language function, https://reference.wolfram.com/language/ref/FactorInteger.html (updated 2007).
✖
Wolfram Research (1988), FactorInteger, Wolfram Language function, https://reference.wolfram.com/language/ref/FactorInteger.html (updated 2007).

Text

Wolfram Research (1988), FactorInteger, Wolfram Language function, https://reference.wolfram.com/language/ref/FactorInteger.html (updated 2007).

✖
Wolfram Research (1988), FactorInteger, Wolfram Language function, https://reference.wolfram.com/language/ref/FactorInteger.html (updated 2007).

CMS

Wolfram Language. 1988. "FactorInteger." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/FactorInteger.html.

✖
Wolfram Language. 1988. "FactorInteger." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/FactorInteger.html.

APA

Wolfram Language. (1988). FactorInteger. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FactorInteger.html

✖
Wolfram Language. (1988). FactorInteger. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FactorInteger.html

BibTeX

@misc{reference.wolfram_2025_factorinteger, author="Wolfram Research", title="{FactorInteger}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/FactorInteger.html}", note=[Accessed: 04-June-2025 ]}

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@misc{reference.wolfram_2025_factorinteger, author="Wolfram Research", title="{FactorInteger}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/FactorInteger.html}", note=[Accessed: 04-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_factorinteger, organization={Wolfram Research}, title={FactorInteger}, year={2007}, url={https://reference.wolfram.com/language/ref/FactorInteger.html}, note=[Accessed: 04-June-2025 ]}

✖
@online{reference.wolfram_2025_factorinteger, organization={Wolfram Research}, title={FactorInteger}, year={2007}, url={https://reference.wolfram.com/language/ref/FactorInteger.html}, note=[Accessed: 04-June-2025 ]}