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Gauss's Cyclotomic Formula


Let p>3 be a prime number, then

 4(x^p-y^p)/(x-y)=R^2(x,y)-(-1)^((p-1)/2)pS^2(x,y),

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

 4Phi_n(z)=A_n^2(z)-(-1)^((n-1)/2)nz^2B_n^2(z),

where A_n(z) and B_n(z) have integer coefficients and are of degree phi(n)/2 and phi(n)/2-2, respectively, with phi(n) the totient function and Phi_n(z) a cyclotomic polynomial. In addition, A_n(z) is symmetric if n is even; otherwise it is antisymmetric. B_n(z) is symmetric in most cases, but it antisymmetric if n is of the form 4k+3 (Riesel 1994, p. 436). The following table gives the first few A_n(z) and B_n(z)s (Riesel 1994, pp. 436-442).

nA_n(z)B_n(z)
52z^2+z+21
72z^3+z^2-z-2z+1
112z^5+z^4-2z^3+2z^2-z-2z^3+1

See also

Aurifeuillean Factorization, Cyclotomic Polynomial, Lucas's Theorem

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References

Gauss, C. F. §356-357 in Untersuchungen über höhere Arithmetik. New York: Chelsea, pp. 425-428 and 467, 1965.Kraitchik, M. Recherches sue la théorie des nombres, tome I. Paris: Gauthier-Villars, pp. 93-129, 1924.Kraitchik, M. Recherches sue la théorie des nombres, tome II. Paris: Gauthier-Villars, pp. 1-5, 1929.Riesel, H. "Gauss's Formula for Cyclotomic Polynomials." In tables at end of Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 436-442, 1994.

Referenced on Wolfram|Alpha

Gauss's Cyclotomic Formula

Cite this as:

Weisstein, Eric W. "Gauss's Cyclotomic Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GausssCyclotomicFormula.html

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