Sum of Cubes of 3 Consecutive Integers which is Square
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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
\(\ds 1^3 + 2^3 + 3^3\) | \(=\) | \(\ds 1 + 8 + 27\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 36\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 6^2\) | ||||||||||||
\(\ds 23^3 + 24^3 + 25^3\) | \(=\) | \(\ds 12 \, 167 + 13 \, 824 + 15 \, 625\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 41 \, 616\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 204^2\) | ||||||||||||
\(\ds \paren {-1}^3 + 0^3 + 1^3\) | \(=\) | \(\ds -1 + 0 + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0^2\) | ||||||||||||
\(\ds 0^3 + 1^3 + 2^3\) | \(=\) | \(\ds 0 + 1 + 8\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3^2\) |
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 right hand side:
\(\ds m^2\) | \(=\) | \(\ds \paren {n - 1}^3 + n^3 + \paren {n + 1}^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n^3 - 3 n^2 + 3 n - 27 + n^3 + n^3 + 3 n^2 + 3 n + 27\) | Cube of Sum, Cube of Difference | |||||||||||
\(\ds \) | \(=\) | \(\ds 3 n^3 + 6 n\) |
Substituting $y = 3 m$ and $x = 3 n$:
\(\ds \paren {\frac y 3}^2\) | \(=\) | \(\ds 3 \paren {\frac x 3}^3 + 6 \paren {\frac x 3}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {y^2} 9\) | \(=\) | \(\ds \frac {x^3} 9 + 2 x\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y^2\) | \(=\) | \(\ds x^3 + 18 x\) |
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.
$\blacksquare$
Also see
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $204$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $204$
- The LMFDB Collaboration, The L-functions and Modular Forms Database, Elliptic Curve 2304/a/2, $2013$ [Online; accessed 31-Mar-2022]