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

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


 * $400 = 1 + 7 + 49 + 343$