Smallest Quadruplet of Consecutive Integers Divisible by Cube

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

\(\ds 22 \, 624\)

\(=\)

\(\ds 2828 \times 2^3\)

\(\ds 22 \, 625\)

\(=\)

\(\ds 181 \times 5^3\)

\(\ds 22 \, 626\)

\(=\)

\(\ds 838 \times 3^3\)

\(\ds 22 \, 627\)

\(=\)

\(\ds 17 \times 11^3\)

This theorem requires a proof. In particular: It remains to be shown there is no smaller such quadruplet. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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): $1375$