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:
- $0, 1, 2, 3, 4$
- $-2, -1, 0, 1, 2$
This sequence is A126203 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
\(\ds 1^3 + 2^3 + 3^3 + 4^3 + 5^3\) | \(=\) | \(\ds 1 + 8 + 27 + 64 + 125\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 225\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 15^2\) | also see Sum of Sequence of Cubes |
\(\ds 25^3 + 26^3 + 27^3 + 28^3 + 29^3\) | \(=\) | \(\ds 15 \, 625 + 17 \, 576 + 19 \, 683 + 21 \, 952 + 24 \, 389\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 99 \, 225\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 315^2\) |
\(\ds 96^3 + 97^3 + 98^3 + 99^3 + 100^3\) | \(=\) | \(\ds 884 \, 736 + 912 \, 673 + 941 \, 192 + 970 \, 299 + 1 \, 000 \, 000\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 \, 708 \, 900\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2170^2\) |
\(\ds 118^3 + 119^3 + 120^3 + 121^3 + 122^3\) | \(=\) | \(\ds 1 \, 643 \, 032 + 1 \, 685 \, 159 + 1 \, 728 \, 000 + 1 \, 771 \, 561 + 1 \, 815 \, 848\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 8 \, 643 \, 600\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2940^2\) |
Then we also have:
\(\ds 0^3 + 1^3 + 2^3 + 3^3 + 4^3\) | \(=\) | \(\ds 0 + 1 + 8 + 27 + 64\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 100\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 10^2\) | also see Sum of Sequence of Cubes |
and finally the degenerate case:
\(\ds \paren {-2}^3 + \paren {-1}^3 + 0^3 + 1^3 + 2^3\) | \(=\) | \(\ds \paren {-8} + \paren {-1} + 0 + 1 + 8\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0^2\) |
Any sequence of $5$ consecutive integers that have cubes that sum to a square would satisfy:
- $m^2 = \paren {n - 2}^3 + \paren {n - 1}^3 + n^3 + \paren {n + 1}^3 + \paren {n + 2}^3$
where $n$ is the middle number of the sequence, with $m, n \in \Z$.
Expanding the right hand side:
\(\ds m^2\) | \(=\) | \(\ds \paren {n - 2}^3 + \paren {n - 1}^3 + n^3 + \paren {n + 1}^3 + \paren {n + 2}^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n^3 - 6 n^2 + 12 n - 8 + n^3 - 3 n^2 + 3 n - 27 + n^3 + n^3 + 3 n^2 + 3 n + 27 + n^3 + 6 n^2 + 12 n + 8\) | Cube of Sum, Cube of Difference | |||||||||||
\(\ds \) | \(=\) | \(\ds 5 n^3 + 30 n\) |
Substituting $y = 5 m$ and $x = 5 n$:
\(\ds \paren {\frac y 5}^2\) | \(=\) | \(\ds 5 \paren {\frac x 5}^3 + 30 \paren {\frac x 5}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {y^2} {25}\) | \(=\) | \(\ds \frac {x^3} {25} + 6 x\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y^2\) | \(=\) | \(\ds x^3 + 150 x\) |
which is an elliptic curve.
According to LMFDB, this elliptic curve has exactly $13$ lattice points:
- $\tuple {0, 0}, \tuple {10, \pm 50}, \tuple {15, \pm 75}, \tuple {24, \pm 132}, \tuple {135, \pm 1575}, \tuple {490, \pm 10 \, 850}, \tuple {600, \pm 14 \, 700}$
which correspond to these values of $n$:
- $0, 2, 3, \dfrac {24} 5, 27, 98, 120$
Note that $\dfrac {24} 5$ is not an integer.
Hence there are no more solutions.
$\blacksquare$
Also see
Historical Note
This result was originally published by Édouard Lucas in $1873$, where he wrote:
- The sum of the cubes of $5$ consecutive numbers is never equal to a square, except for the solutions of which the middle numbers are $2$, $3$, $27$, $98$ or $120$.
Subsequently it was reported by Leonard Eugene Dickson, who made a mistake by omitting the $27$.
The erroneous sequence is still mistakenly published on occasion.
Sources
- 1873: E. Lucas: Recherches sur l'analyse indéterminée (Bull. Soc. d'Emulation du Departement de l'Allier Vol. 12: p. 532)
- 1920: Leonard Eugene Dickson: History of the Theory of Numbers: Volume $\text { II }$: Chapter $21$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $118$
- The LMFDB Collaboration, The L-functions and Modular Forms Database, Elliptic Curve 57600/bt/2, $2013$ [Online; accessed 31-Mar-2022]