Property of 490,689

Theorem
The number $490 \, 689$ can be expressed as the sum of $3$ cubes in $2$ different ways:
 * $490 \, 689 = 4^3 + 60^3 + 65^3 = 8^3 + 25^3 \times 78^3$

while at the same time the products of the contributory cube roots of each of those $2$ ways are equal:
 * $4 \times 60 \times 65 = 8 \times 25 \times 78$

Proof
Then: