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
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1225$