Sum of Cubes of 5 Consecutive Integers which is Square

Theorem
The following sequences of $5$ consecutive (strictly) positive integers have cubes that sum to squares:


 * $1, 2, 3, 4, 5$


 * $25, 26, 27, 28, 29$


 * $96, 97, 98, 99, 100$


 * $118, 119, 120, 121, 122$

No other such sequence of $5$ consecutive positive integers has the same property.

However, if we allow sequences containing zero and negative integers, we also have:


 * $0, 1, 2, 3, 4$


 * $-2, -1, 0, 1, 2$

Proof
Then we also have:

and finally the degenerate case: