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CompositeQ
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CompositeQ

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yields True if n is a composite number, and yields False otherwise.

Details and Options

  • CompositeQ is typically used to test whether an integer is a composite number.
  • A composite number is a positive number that is the product of two integers other than 1.
  • CompositeQ[n] returns False unless n is manifestly a composite number.
  • For negative integer n, CompositeQ[n] is effectively equivalent to CompositeQ[-n].
  • With the setting GaussianIntegers->True, CompositeQ determines whether a number is a composite number over Gaussian integers.
  • CompositeQ[m+In] automatically works over Gaussian integers.

Examples

open allclose all

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

Test whether a number is composite:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-mgv65a
Out[1]=1

The number is not composite:

In[1]:=1
✖
https://wolfram.com/xid/0bdo3ycm7-tsbby3
Out[1]=1

Scope  (4)Survey of the scope of standard use cases

CompositeQ works over integers:

In[3]:=3
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https://wolfram.com/xid/0bdo3ycm7-mv5gft
Out[3]=3

Gaussian integers:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-srsb6y
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bdo3ycm7-nqms7k
Out[2]=2

Test for large integers:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-845fn4
Out[1]=1

CompositeQ threads over lists:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-crn
Out[1]=1

Options  (1)Common values & functionality for each option

GaussianIntegers  (1)

Test whether is composite over integers:

In[10]:=10
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https://wolfram.com/xid/0bdo3ycm7-10e1vr
Out[10]=10

Gaussian integers:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-s7a4h8
Out[1]=1

Applications  (9)Sample problems that can be solved with this function

Basic Applications  (3)

Highlight composite numbers:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-bfkjsd
Out[1]=1

Generate the composite number:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-ct38cw
In[2]:=2
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https://wolfram.com/xid/0bdo3ycm7-ljas3i
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0bdo3ycm7-gfuszz
Out[3]=3

Generate random composite numbers:

In[4]:=4
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https://wolfram.com/xid/0bdo3ycm7-wq2i48
In[5]:=5
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https://wolfram.com/xid/0bdo3ycm7-ibhzx
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0bdo3ycm7-ncv8w
Out[6]=6

The distribution of Gaussian composite numbers:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-b620o8
Out[1]=1

Number Theory  (6)

Recognize Sierpiński numbers k where is always composite:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-uvt145
Out[2]=2

Recognize powerful numbers n whose prime factors are all repeated:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-3x7iu
In[2]:=2
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https://wolfram.com/xid/0bdo3ycm7-bps1v3
Out[2]=2

All perfect powers are powerful numbers:

In[3]:=3
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https://wolfram.com/xid/0bdo3ycm7-g4cjq
Out[3]=3

Recognize base b pseudoprimes, composite numbers n such that :

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-73ghej

Find all base pseudoprimes below :

In[2]:=2
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https://wolfram.com/xid/0bdo3ycm7-ddotbl
Out[2]=2

Find all base pseudoprimes below :

In[3]:=3
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https://wolfram.com/xid/0bdo3ycm7-v2utbo
Out[3]=3

Find large composite numbers of the form :

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-c3ghq8
Out[1]=1

The distribution of composite numbers over integers:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-hi0e4n
In[2]:=2
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https://wolfram.com/xid/0bdo3ycm7-iul1cx

Plot the distribution:

In[3]:=3
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https://wolfram.com/xid/0bdo3ycm7-d4lpp
Out[3]=3

The distribution of composite numbers over the Gaussian integers:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-9om2p7
In[2]:=2
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https://wolfram.com/xid/0bdo3ycm7-1ullvw

Plot the distribution:

In[3]:=3
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https://wolfram.com/xid/0bdo3ycm7-jrlfrw
Out[3]=3

Properties & Relations  (6)Properties of the function, and connections to other functions

Primes represents the domain of all prime numbers:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-d89n7a
Out[1]=1

PrimeQ gives False for all composite numbers:

In[4]:=4
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https://wolfram.com/xid/0bdo3ycm7-cocnkk
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0bdo3ycm7-dz14y
Out[5]=5

CompositeQ gives False for all primes:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-4sc00
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bdo3ycm7-jf1u0m
Out[2]=2

Composite numbers cannot be a MersennePrimeExponent:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-blul38
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bdo3ycm7-c5un5t
Out[2]=2

Composite numbers have at least two prime factors including multiplicities:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-z74y0b
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bdo3ycm7-krrkpj
Out[2]=2

Composite numbers that are the power of a prime number have exactly divisors:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-e8dstd
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bdo3ycm7-fryb42
Out[2]=2

Neat Examples  (2)Surprising or curious use cases

Plot composite numbers that are the sum of three squares:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-h7zwj2
Out[1]=1

Plot the Ulam spiral where numbers are colored based on their compositeness:

In[1]:=1
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https://wolfram.com/xid/0bdo3ycm7-e8n2ht
In[2]:=2
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https://wolfram.com/xid/0bdo3ycm7-qhm0kv
Out[2]=2
Wolfram Research (2014), CompositeQ, Wolfram Language function, https://reference.wolfram.com/language/ref/CompositeQ.html.
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Wolfram Research (2014), CompositeQ, Wolfram Language function, https://reference.wolfram.com/language/ref/CompositeQ.html.

Text

Wolfram Research (2014), CompositeQ, Wolfram Language function, https://reference.wolfram.com/language/ref/CompositeQ.html.

✖
Wolfram Research (2014), CompositeQ, Wolfram Language function, https://reference.wolfram.com/language/ref/CompositeQ.html.

CMS

Wolfram Language. 2014. "CompositeQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CompositeQ.html.

✖
Wolfram Language. 2014. "CompositeQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CompositeQ.html.

APA

Wolfram Language. (2014). CompositeQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CompositeQ.html

✖
Wolfram Language. (2014). CompositeQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CompositeQ.html

BibTeX

@misc{reference.wolfram_2025_compositeq, author="Wolfram Research", title="{CompositeQ}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/CompositeQ.html}", note=[Accessed: 02-May-2025 ]}

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@misc{reference.wolfram_2025_compositeq, author="Wolfram Research", title="{CompositeQ}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/CompositeQ.html}", note=[Accessed: 02-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_compositeq, organization={Wolfram Research}, title={CompositeQ}, year={2014}, url={https://reference.wolfram.com/language/ref/CompositeQ.html}, note=[Accessed: 02-May-2025 ]}

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@online{reference.wolfram_2025_compositeq, organization={Wolfram Research}, title={CompositeQ}, year={2014}, url={https://reference.wolfram.com/language/ref/CompositeQ.html}, note=[Accessed: 02-May-2025 ]}