Sum of Cubes of 5 Consecutive Integers which is Square

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


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


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


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

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