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 |