Let p > 3 be a prime number, then
where R(x,y) and S(x,y) are homogeneous polynomials in x and y with integer coefficients. Gauss (1965, p. 467) gives the coefficients of R and S up to p = 23.
Kraitchik (1924) generalized Gauss's formula to odd squarefree integers n > 3. Then Gauss's formula can be written in the slightly simpler form
where
and 
have integer coefficients and are of degree 
and 
, respectively, with 
the totient function and 
a cyclotomic polynomial. In addition, is symmetric if n is even;otherwise it is antisymmetric. is symmetric in most cases, but it antisymmetric if n is of the form 
(Riesel 1994, p. 436). The following table gives the first few and s (Riesel 1994, pp. 436-442).
[left]n[/left]
[left]5[/left]
[left]7[/left]
[left]11[/left]
where R(x,y) and S(x,y) are homogeneous polynomials in x and y with integer coefficients. Gauss (1965, p. 467) gives the coefficients of R and S up to p = 23.
Kraitchik (1924) generalized Gauss's formula to odd squarefree integers n > 3. Then Gauss's formula can be written in the slightly simpler form
where
and have integer coefficients and are of degree and , respectively, with the totient function and a cyclotomic polynomial. In addition, is symmetric if n is even;otherwise it is antisymmetric. is symmetric in most cases, but it antisymmetric if n is of the form (Riesel 1994, p. 436). The following table gives the first few and s (Riesel 1994, pp. 436-442). [left]n[/left]
[left]5[/left]
1
[left]7[/left]
[left]11[/left]
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