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KroneckerSymbol
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KroneckerSymbol

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gives the Kronecker symbol .

Details

  • KroneckerSymbol is also known as the Jacobi symbol or Legendre symbol.
  • Integer mathematical function, suitable for both symbolic and numerical manipulation.
  • KroneckerSymbol[n,1] gives 1.
  • KroneckerSymbol[n,-1] gives 1 whenever n is non-negative and otherwise.
  • For a number with a unit and primes, TemplateBox[{n, m}, KroneckerSymbol] returns TemplateBox[{n, u}, KroneckerSymbol] TemplateBox[{n, {p, _, 1}}, KroneckerSymbol] ...TemplateBox[{n, {p, _, l}}, KroneckerSymbol].

Examples

open allclose all

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

Compute Kronecker symbols:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-psrsi
Out[1]=1
In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-9jh4d
Out[1]=1

Plot the KroneckerSymbol sequence with respect to the second argument:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-ihvhm2
Out[1]=1

Scope  (9)Survey of the scope of standard use cases

Numerical Evaluation  (3)

KroneckerSymbol works over integers:

In[3]:=3
✖
https://wolfram.com/xid/0bto00wuqweq-boeak4
Out[3]=3

Compute for large arguments:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-i1usxz
Out[1]=1

KroneckerSymbol threads elementwise over lists:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-dkvvun
Out[1]=1

Symbolic Manipulation  (6)

TraditionalForm formatting:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-3wfjxa

Reduce expressions:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-84mt0
Out[1]=1

Solve equations:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-p4vq2l
Out[1]=1

Use KroneckerSymbol in a sum:

In[1]:=1
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https://wolfram.com/xid/0bto00wuqweq-gtwzef
Out[1]=1

Recurrence equation:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-041jh2
Out[1]=1

Generating function:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-i66f2g
Out[1]=1

Applications  (11)Sample problems that can be solved with this function

Basic Applications  (2)

Table of values of the Kronecker symbol with n, m up to 10:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-eo6feb
Out[1]=1

Plot the nontrivial values of the Kronecker symbol:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-cqkxxi
Out[1]=1

Number Theory  (9)

For congruent integers m and n modulo p, KroneckerSymbol[m,p]==KroneckerSymbol[n,p]:

In[3]:=3
✖
https://wolfram.com/xid/0bto00wuqweq-dbmikw
Out[3]=3
In[2]:=2
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https://wolfram.com/xid/0bto00wuqweq-m7kvl2
Out[2]=2

Find Euler–Jacobi pseudoprimes to base : [more info]

In[1]:=1
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https://wolfram.com/xid/0bto00wuqweq-cojb1p
In[2]:=2
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https://wolfram.com/xid/0bto00wuqweq-h66s3f
Out[2]=2

The law of quadratic reciprocity for distinct primes n and m :

In[1]:=1
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https://wolfram.com/xid/0bto00wuqweq-ercgj8
In[2]:=2
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https://wolfram.com/xid/0bto00wuqweq-fl0dql
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bto00wuqweq-ugtc5
Out[3]=3

Construct eigenvectors of the discrete Fourier transform:

In[1]:=1
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https://wolfram.com/xid/0bto00wuqweq-fmmr1y
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bto00wuqweq-jg44or
Out[2]=2

Evaluate Gauss sums in closed form:

In[1]:=1
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https://wolfram.com/xid/0bto00wuqweq-nefum0
In[2]:=2
✖
https://wolfram.com/xid/0bto00wuqweq-q0jkvn
Out[2]=2

The congruence equation has a solution if KroneckerSymbol[a,p] == 1:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-dfgeyz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bto00wuqweq-hs7spj
Out[2]=2

KroneckerSymbol[n,k] is a real DirichletCharacter modulo k for odd integers k:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-k0h94z
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bto00wuqweq-ikrxpv
Out[2]=2

A real primitive character χ modulo k can be written in terms of KroneckerSymbol[χ[-1]k,n]:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-nm1zl
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bto00wuqweq-ebfxtg
Out[2]=2

Nonprimitive real characters can be written in terms of KroneckerSymbol at integers coprime to k:

In[3]:=3
✖
https://wolfram.com/xid/0bto00wuqweq-ftu2et
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bto00wuqweq-fzgbzy
Out[4]=4

KroneckerSymbol is the generalization of the Jacobi symbol for all integers:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-fmnnm6
In[2]:=2
✖
https://wolfram.com/xid/0bto00wuqweq-49qyus
Out[2]=2

Properties & Relations  (5)Properties of the function, and connections to other functions

KroneckerSymbol gives for non-coprime integers:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-bp8jdm
Out[1]=1
In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-lq2898
Out[1]=1

KroneckerSymbol is a completely multiplicative function for each argument:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-jrnqi6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bto00wuqweq-biwgxu
Out[2]=2

The law of quadratic reciprocity for distinct primes n and m :

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-mwhaay
In[2]:=2
✖
https://wolfram.com/xid/0bto00wuqweq-gdpsdl
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bto00wuqweq-bisjpg
Out[3]=3

Use KroneckerSymbol to compute real DirichletCharacter modulo k for odd integers k:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-p85kl
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bto00wuqweq-c2oc3z
Out[2]=2

Check that the following relation holds for any odd integer:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-fc0vsk
Out[1]=1

Neat Examples  (4)Surprising or curious use cases

The array plot of KroneckerSymbol:

In[4]:=4
✖
https://wolfram.com/xid/0bto00wuqweq-hdfirj
Out[4]=4

Plot the arguments of the Fourier transform of KroneckerSymbol:

In[1]:=1
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https://wolfram.com/xid/0bto00wuqweq-d7etv5
Out[1]=1

Successive differences of KroneckerSymbol modulo 2:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-j89uev
Out[1]=1

Plot the Ulam spiral of KroneckerSymbol:

In[1]:=1
✖
https://wolfram.com/xid/0bto00wuqweq-xloq7k
In[2]:=2
✖
https://wolfram.com/xid/0bto00wuqweq-gwe44e
Out[2]=2
Wolfram Research (2007), KroneckerSymbol, Wolfram Language function, https://reference.wolfram.com/language/ref/KroneckerSymbol.html.
✖
Wolfram Research (2007), KroneckerSymbol, Wolfram Language function, https://reference.wolfram.com/language/ref/KroneckerSymbol.html.

Text

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

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

CMS

Wolfram Language. 2007. "KroneckerSymbol." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/KroneckerSymbol.html.

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

APA

Wolfram Language. (2007). KroneckerSymbol. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/KroneckerSymbol.html

✖
Wolfram Language. (2007). KroneckerSymbol. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/KroneckerSymbol.html

BibTeX

@misc{reference.wolfram_2025_kroneckersymbol, author="Wolfram Research", title="{KroneckerSymbol}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/KroneckerSymbol.html}", note=[Accessed: 03-May-2025 ]}

✖
@misc{reference.wolfram_2025_kroneckersymbol, author="Wolfram Research", title="{KroneckerSymbol}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/KroneckerSymbol.html}", note=[Accessed: 03-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_kroneckersymbol, organization={Wolfram Research}, title={KroneckerSymbol}, year={2007}, url={https://reference.wolfram.com/language/ref/KroneckerSymbol.html}, note=[Accessed: 03-May-2025 ]}

✖
@online{reference.wolfram_2025_kroneckersymbol, organization={Wolfram Research}, title={KroneckerSymbol}, year={2007}, url={https://reference.wolfram.com/language/ref/KroneckerSymbol.html}, note=[Accessed: 03-May-2025 ]}