Gauss's Formula

$$4{x^p-y^p\over x-y} =R^2(x,y)-(-1)^{(p-1)/2} pS^2(x,y),$$

where $R$ and $S$ are Homogeneous Polynomials in $x$ and $y$ with integral Coefficients.

See also Aurifeuillean Factorization, Gauss's Backward Formula, Gauss's Forward Formula

References

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 105, 1993.