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PrimeQ
✖
PrimeQ

✖
PrimeQ[n]

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

Details and Options

  • PrimeQ is typically used to test whether an integer is a prime number.
  • A prime number is a positive integer that has no divisors other than 1 and itself.
  • PrimeQ[n] returns False unless n is manifestly a prime number.
  • For negative integer n, PrimeQ[n] is effectively equivalent to PrimeQ[-n].
  • With the setting GaussianIntegers->True, PrimeQ determines whether a number is a Gaussian prime.
  • PrimeQ[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 prime:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-e2s
Out[1]=1

The number 4 is not prime:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-n7e
Out[1]=1

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

PrimeQ works over integers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-u0o0br
Out[1]=1

Gaussian integers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-wo26j4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-nfavki
Out[2]=2

Test for large integers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-845fn4
Out[1]=1

PrimeQ threads over lists:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-eulagn
Out[1]=1

Options  (1)Common values & functionality for each option

GaussianIntegers  (1)

Test whether 5 is prime over integers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-10e1vr
Out[1]=1

Gaussian integers:

In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-s7a4h8
Out[2]=2

Applications  (22)Sample problems that can be solved with this function

Basic Applications  (5)

Highlight prime numbers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-mi7qim
Out[1]=1

Generate prime number:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-m7tjfe
Out[1]=1

Generate random prime numbers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-bk64lk
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-c1yx4g
Out[2]=2

The distribution of Gaussian primes:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-b620o8
Out[1]=1

Find the first few prime powers that are not prime:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-bmhtty
Out[1]=1

Special Sequences  (11)

Plot Gaussian primes:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-ej8xp5
In[3]:=3
✖
https://wolfram.com/xid/0wgi29f-6y3qa
Out[3]=3

Eisenstein integers are complex numbers of the form where a and b are integers and ω is the cube root of unity :

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-mhbezn
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-yfolsm

Check wether an Eisenstein integer is prime:

In[3]:=3
✖
https://wolfram.com/xid/0wgi29f-iy83y0
In[4]:=4
✖
https://wolfram.com/xid/0wgi29f-9cenw5
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0wgi29f-khllhf
Out[5]=5

Plot Eisenstein primes:

In[6]:=6
✖
https://wolfram.com/xid/0wgi29f-dxrxgc
In[7]:=7
✖
https://wolfram.com/xid/0wgi29f-qz4c9y
Out[7]=7

The quadratic polynomial is prime for :

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-dj4898
Out[1]=1

Recognize Fermat primes, prime numbers of the form :

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-mj1gl7
Out[1]=1

The number is not a Fermat prime:

In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-jp8z0t
Out[2]=2

Recognize Carmichael numbers, composite numbers n that satisfy mod for all integers b that are relatively prime to n:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-mjbk4f

The number 1729 is a Carmichael number; 1310 is not:

In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-8e5nta
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0wgi29f-pwumyg
Out[3]=3

Recognize Wieferich primes, prime numbers p such that divides :

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-hndplo
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-len25z
Out[2]=2

There are only two known Wieferich primes:

In[3]:=3
✖
https://wolfram.com/xid/0wgi29f-3y6osb
Out[3]=3

Recognize Gaussian Mersenne primes, prime numbers n such that is a Gaussian prime:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-czd2jj
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-b7rsdv
Out[2]=2

Let be all numbers of the form :

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-y7zsqg

Check that the product of two numbers is still in :

In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-61uqh7
Out[2]=2

Recognize Hilbert primes, prime numbers that have no divisors in other than 1 and itself:

In[3]:=3
✖
https://wolfram.com/xid/0wgi29f-4pv3e7

Find the first 10 Hilbert primes:

In[4]:=4
✖
https://wolfram.com/xid/0wgi29f-2swwye
Out[4]=4

Test whether or not the first 47 Mersenne prime exponents are prime:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-romquq
Out[1]=1

Find twin primes:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-ql3hbb
Out[1]=1

Find Mersenne prime exponents:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-n63f3p
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-d69xgz
Out[2]=2

Number Theory  (6)

Find numbers that are prime over Gaussian integers and integers:

In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-ttmtf
Out[2]=2

They are congruent to 3 mod 4:

In[5]:=5
✖
https://wolfram.com/xid/0wgi29f-j1ne11
Out[5]=5

These numbers cannot be written as the sum of two squares:

In[6]:=6
✖
https://wolfram.com/xid/0wgi29f-dn44j6
Out[6]=6

Find numbers that are composite over Gaussian integers but prime over integers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-9f178g
Out[1]=1

All of them except for 2 are congruent to 1 mod 4:

In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-nllsgi
Out[2]=2

These numbers can be written as the sum of two squares in 8 ways:

In[3]:=3
✖
https://wolfram.com/xid/0wgi29f-rd3ngt
Out[3]=3

Plot the difference between two consecutive primes:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-jmvbwl
Out[1]=1

The infinite sum of reciprocals of prime powers that are not prime converges:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-lu7z7a
Out[1]=1

The distribution of primes over integers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-hi0e4n
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-iul1cx

Plot the distribution:

In[3]:=3
✖
https://wolfram.com/xid/0wgi29f-d4lpp
Out[3]=3

The distribution of Gaussian primes over Gaussian integers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-9om2p7
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-1ullvw

Plot the distribution:

In[3]:=3
✖
https://wolfram.com/xid/0wgi29f-jrlfrw
Out[3]=3

Properties & Relations  (22)Properties of the function, and connections to other functions

Primes represents the domain of all prime numbers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-2wjcms
Out[1]=1

Prime gives prime number:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-bwxgxm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-m3768e
Out[2]=2

RandomPrime generates random prime numbers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-b2r76k
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-l1e1p
Out[2]=2

PrimePowerQ gives True for all prime numbers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-g0tfh2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-b2ds79
Out[2]=2

Primes that are congruent to 1 mod 4 are not prime powers in the Gaussian integers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-i0nw8j
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-7k9zwv
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0wgi29f-0ahika
Out[3]=3

Prime powers are divisible by exactly one prime number:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-j0ie3
Out[1]=1
In[5]:=5
✖
https://wolfram.com/xid/0wgi29f-dfj2iy
Out[5]=5

The only divisors of a prime number p is 1 and p:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-bj49w7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-ioq1b5
Out[2]=2

The only even prime number is 2:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-dz613q
Out[1]=1

PrimeQ gives False for all composite numbers:

In[4]:=4
✖
https://wolfram.com/xid/0wgi29f-cocnkk
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0wgi29f-dz14y
Out[5]=5

CompositeQ gives False for all primes:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-4sc00
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-jf1u0m
Out[2]=2

Every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-ky3hs2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-e7lr3s
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0wgi29f-fjuubb
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0wgi29f-d2b1t6
Out[4]=4

The GCD of two prime numbers is 1; consequently, two prime numbers are relatively prime:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-fd85es
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-cq1ayy
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0wgi29f-cwgicu
Out[3]=3

The LCM for prime numbers is their product:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-c11ruk
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-b6ga13
Out[2]=2

The sum of the prime divisors of a prime number returns the original number:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-km73bd
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-7qsbw7
Out[2]=2

Prime numbers of the form , where the exponent p is also prime, are called Mersenne primes:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-djva1f
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-dxitd8
Out[2]=2

MersennePrimeExponents are prime numbers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-i7supp
Out[1]=1

Use FactorInteger to find all prime divisors of a number:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-pefmwt
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-ona38t
Out[2]=2

PrimeOmega returns 1 for prime numbers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-ok5wcm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-do9kut
Out[2]=2

PrimePi gives the number of primes:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-7ikr8v
Out[1]=1

The number of prime numbers up to 1000:

In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-co0tmt
Out[2]=2

PrimeNu counts the number of prime divisors of a number:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-u4r83x
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-c3zirp
Out[2]=2

Simplify expressions containing prime numbers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-7w1yvx
Out[1]=1

Solve over Primes:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-vjjf3s
Out[1]=1

Interactive Examples  (1)Examples with interactive outputs

The polar plot of primes:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-p2ugis
Out[1]=1

Neat Examples  (3)Surprising or curious use cases

Visualize when is divisible by primes. Each row of dots corresponds to the divisors of , which are labeled along the horizontal axis:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-eq61yv
Out[1]=1

Plot the prime numbers that are the sum of three squares:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-l9t
Out[1]=1

Plot the Ulam spiral of prime numbers:

In[1]:=1
✖
https://wolfram.com/xid/0wgi29f-l5q7y3
In[2]:=2
✖
https://wolfram.com/xid/0wgi29f-vcxncv
Out[2]=2
Wolfram Research (1988), PrimeQ, Wolfram Language function, https://reference.wolfram.com/language/ref/PrimeQ.html (updated 2003).
✖
Wolfram Research (1988), PrimeQ, Wolfram Language function, https://reference.wolfram.com/language/ref/PrimeQ.html (updated 2003).

Text

Wolfram Research (1988), PrimeQ, Wolfram Language function, https://reference.wolfram.com/language/ref/PrimeQ.html (updated 2003).

✖
Wolfram Research (1988), PrimeQ, Wolfram Language function, https://reference.wolfram.com/language/ref/PrimeQ.html (updated 2003).

CMS

Wolfram Language. 1988. "PrimeQ." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2003. https://reference.wolfram.com/language/ref/PrimeQ.html.

✖
Wolfram Language. 1988. "PrimeQ." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2003. https://reference.wolfram.com/language/ref/PrimeQ.html.

APA

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

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

BibTeX

@misc{reference.wolfram_2025_primeq, author="Wolfram Research", title="{PrimeQ}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/PrimeQ.html}", note=[Accessed: 18-May-2025 ]}

✖
@misc{reference.wolfram_2025_primeq, author="Wolfram Research", title="{PrimeQ}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/PrimeQ.html}", note=[Accessed: 18-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_primeq, organization={Wolfram Research}, title={PrimeQ}, year={2003}, url={https://reference.wolfram.com/language/ref/PrimeQ.html}, note=[Accessed: 18-May-2025 ]}

✖
@online{reference.wolfram_2025_primeq, organization={Wolfram Research}, title={PrimeQ}, year={2003}, url={https://reference.wolfram.com/language/ref/PrimeQ.html}, note=[Accessed: 18-May-2025 ]}