Squares which are Difference between Two Cubes

Theorem

$169$ is the smallest square number which is the difference between two cubes:

$169 = 8^3 - 7^3$

This article is complete as far as it goes, but it could do with expansion.
In particular: Add the rest of the sequence, having found out what they are.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.
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 {{Expand}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Proof

$\ds 8^3 - 7^3$

$=$

$\ds 512 - 343$

$\ds $

$=$

$\ds 169$

$\ds $

$=$

$\ds 13^2$

This theorem requires a proof.
In particular: Establish that this is the smallest such.
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.

Sources

1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $169$