Ford's Theorem

Let $a$, $b$, and $k$ be Integers with $k\geq 1$. For $j=0$, 1, 2, let

$$S_j\equiv \sum_{i\equiv j\ \left({{\rm mod\ } {3}}\right)} (-1)^j{k\choose i}a^{k-i}b^i.$$

Then

$$2(a^2+ab+b^2)^{2k}=(S_0-S_1)^4+(S_1-S_2)^4+(S_2-S_0)^4.$$

See also Bhargava's Theorem, Diophantine Equation--Quartic

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 100-101, 1994.