Numbers whose Cube equals Sum of Sequence of that many Squares/Examples/99,359

Example of Number whose Cube equals Sum of Sequence of that many Squares

 * $\displaystyle 99 \, 359^3 = \sum_{k \mathop = 1}^{99 \, 359} \paren {45 \, 449 + k}^2$

Proof
From Numbers whose Cube equals Sum of Sequence of that many Squares:
 * $\displaystyle m^3 = \sum_{k \mathop = 1}^m \paren {n + k}^2$

for $m = 99 \, 359$ and for $n$ given by:


 * $n = \dfrac {m + 1} 2 \pm \dfrac 1 6 \sqrt {33 m^2 + 3}$

So in this instance:

This gives:
 * $n = 45 \, 449, n = -144 \, 809$

which leads to the two solutions: