# Smallest Quadruplet of Consecutive Integers Divisible by Cube

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## Contents

## Theorem

The smallest sequence of quadruplets of consecutive integers each of which is divisible by a cube greater than $1$ is:

- $\tuple {22 \, 624, 22 \, 625, 22 \, 626, 22 \, 627}$

## Proof

\(\displaystyle 22 \, 624\) | \(=\) | \(\displaystyle 2828 \times 2^3\) | |||||||||||

\(\displaystyle 22 \, 625\) | \(=\) | \(\displaystyle 181 \times 5^3\) | |||||||||||

\(\displaystyle 22 \, 626\) | \(=\) | \(\displaystyle 838 \times 3^3\) | |||||||||||

\(\displaystyle 22 \, 627\) | \(=\) | \(\displaystyle 17 \times 11^3\) |

## Historical Note

This result is reported by David Wells in his *Curious and Interesting Numbers, 2nd ed.* of $1997$ as the work of Stephane Vandemergel, but details are lacking.

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $1375$