Let 
be a prime number, then
 |
where 
and 
are homogeneous polynomials in 
and 
with integer coefficients.
Gauss (1965, p. 467) gives the coefficients of 
and 
up to 
.
Kraitchik (1924) generalized Gauss's formula to odd squarefree integers 
.
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 
is even; otherwise it is antisymmetric.
is symmetric in most cases, but it antisymmetric if 
is of the form 
(Riesel 1994, p. 436). The following table gives the
first few 
and 
s
(Riesel 1994, pp. 436-442).
 |  |  |
| 5 |  | 1 |
| 7 |  |  |
| 11 |  |  |
