Square Numbers which are Sum of Sequence of Odd Cubes/Mistake

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Source Work

1997: David Wells: Curious and Interesting Numbers (2nd ed.):

The Dictionary
$1225$


Mistake

$35^2 = 1^3 + 3^3 + 5^3 + 7^3 + 9^3$. The next such sum is $1^3 + \ldots 29^3$.


Correction

This is wrong, as can be seen by the fact that:

$1^3 + \cdots + 29^3 = 101 \, 025$

which is not a square number.

In fact, the next such sum is $1^3 + 3^3 + 5^3 + \dotsb + 55^3 + 57^3$.

That is:

$1^3 + \paren {2 \times 2 - 1}^3 + \paren {2 \times 3 - 1}^3 + \dotsb + \paren {2 \times 28 - 1}^3 + \paren {2 \times 29 - 1}^3$


What David Wells seems to have done (and is seems to be a common confusion) is confuse:

$\ds \sum_{j \mathop = 1}^{29} \paren {2 j - 1}^3$

with:

$1^3 + 3^3 + \dotsb + 27^3 + 29^3$

which is not the same thing at all.


Sources