Sum of Cubes of 3 Consecutive Integers which is Square

Theorem
The following sequences of $3$ consecutive (strictly) positive integers have cubes that sum to a square:


 * $1, 2, 3$


 * $23, 24, 25$

No other such sequence of $3$ consecutive positive integers has the same property.

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


 * $-1, 0, 1$


 * $0, 1, 2$

Proof
Any sequence of $3$ consecutive integers that have cubes that sum to a square would satisfy:
 * $m^2 = \paren {n - 1}^3 + n^3 + \paren {n + 1}^3$

where $n$ is the middle number of the sequence, with $m, n \in \Z$.

Expanding the :

Substituting $y = 3 m$ and $x = 3 n$:

which is an elliptic curve.

According to LMFDB, this elliptic curve has exactly $7$ lattice points:
 * $\tuple {0, 0}, \tuple {3, \pm 9}, \tuple {6, \pm 18}, \tuple {72, \pm 612}$

which correspond to these values of $n$:
 * $0, 1, 2, 24$

Hence there are no more solutions.

Also see

 * Sum of Cubes of 5 Consecutive Integers which is Square