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

$121 = 1 + 3 + 9 + 27 + 81$
$400 = 1 + 7 + 49 + 343$


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In particular: It remains to be shown that these are the only such square numbers.
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Sources

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