Liouville Polynomial Identity

$\quad +(x_1+x_4)^4+(x_2+x_4)^4+(x_3+x_4)^4+(x_1-x_2)^4$
$\quad +(x_1-x_3)^4+(x_2-x_3)^4+(x_1-x_4)^4+(x_2-x_4)^4 +(x_3-x_4)^4.$

This is proven in Rademacher and Toeplitz (1957).

See also Waring's Problem

References

Rademacher, H. and Toeplitz, O. The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, pp. 55-56, 1957.