Square Numbers which are Sum of Consecutive Powers

Theorem

The only two square numbers which are the sum of consecutive powers of a positive integer are $121$ and $400$:

$121 = 3^0 + 3^1 + 3^2 + 3^3 + 3^4 = 11^2$

$400 = 7^0 + 7^1 + 7^2 + 7^3 = 20^2$

Proof

$121 = 1 + 3 + 9 + 27 + 81$

$400 = 1 + 7 + 49 + 343$

This theorem requires a proof. In particular: It remains to be shown that these are the only such square numbers. 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

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $121$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $121$