References
129
is the prime factorization of n and
a
p i
, and the Legendre symbol is defined as
a
p
=
⎧
⎨
⎩
0 if a ≡ 0 (mod p)
1 if a is a quadratic residue modulo p and a ≡ 0 (mod p)
−1 if a is a quadratic non-residue modulo p
,
(9.13)
and in addition,
−1
p
=
1 if p ≡ 1 (mod 4)
−1 if p ≡ 3 (mod 4)
(9.14)
2
p
=
1 if p ≡ 1, 7 (mod 8)
−1 if p ≡ 3, 5 (mod 8)
.
(9.15)
The Jacobi symbol can be computed with the program jacobisymbol. The
syntax is >> js = jacobisymbol(a, n). The input arguments “a, n” are the
variables of the Jacobi symbol
a
n
, and the output variable “js” is the value of this
Jacobi symbol.
References
1. Johansson, F.: Numerical evaluation of elliptic functions, elliptic integrals and modular forms
(2018). https://arxiv.org/abs/1806.06725
2. Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W.,
Miller, B.R., Sounders, B.V. (eds.): NIST Digital Library of Mathematical Functions (2017).
http://dlmf.nist.gov. Rel. 1.0.17
3. Rademacher, H.: Über die Transformation der Logarithmen der Thetafunktionen. Math. Ann.
168, 142 (1967)
4. Wikipedia: Dedekind eta function (2019). https://en.wikipedia.org/wiki/Dedekind_eta_function
5. Wikipedia: Jacobi symbol (2019). https://en.wikipedia.org/wiki/Jacobi_fsymbol
129
is the prime factorization of n and
a
p i
, and the Legendre symbol is defined as
a
p
=
⎧
⎨
⎩
0 if a ≡ 0 (mod p)
1 if a is a quadratic residue modulo p and a ≡ 0 (mod p)
−1 if a is a quadratic non-residue modulo p
,
(9.13)
and in addition,
−1
p
=
1 if p ≡ 1 (mod 4)
−1 if p ≡ 3 (mod 4)
(9.14)
2
p
=
1 if p ≡ 1, 7 (mod 8)
−1 if p ≡ 3, 5 (mod 8)
.
(9.15)
The Jacobi symbol can be computed with the program jacobisymbol. The
syntax is >> js = jacobisymbol(a, n). The input arguments “a, n” are the
variables of the Jacobi symbol
a
n
, and the output variable “js” is the value of this
Jacobi symbol.
References
1. Johansson, F.: Numerical evaluation of elliptic functions, elliptic integrals and modular forms
(2018). https://arxiv.org/abs/1806.06725
2. Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W.,
Miller, B.R., Sounders, B.V. (eds.): NIST Digital Library of Mathematical Functions (2017).
http://dlmf.nist.gov. Rel. 1.0.17
3. Rademacher, H.: Über die Transformation der Logarithmen der Thetafunktionen. Math. Ann.
168, 142 (1967)
4. Wikipedia: Dedekind eta function (2019). https://en.wikipedia.org/wiki/Dedekind_eta_function
5. Wikipedia: Jacobi symbol (2019). https://en.wikipedia.org/wiki/Jacobi_fsymbol