# Cube as Difference between Two Squares

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## Theorem

A cube number can be expressed as the difference between two squares.

## Proof

\(\displaystyle n^3\) | \(=\) | \(\displaystyle \sum_{k \mathop = 1}^n k^3 - \sum_{k \mathop = 1}^{n - 1} k^3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {\frac {n^2 \paren {n + 1}^2} 4} - \paren {\frac {\paren {n - 1}^2 n^2} 4}\) | Sum of Sequence of Cubes | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {n^2 \paren {\paren {n + 1}^2 - \paren {n - 1}^2} } 4\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {\frac {n \paren {n + 1} } 2}^2 - \paren {\frac {n \paren {n - 1} } 2}^2\) |

$\blacksquare$

## Sources

- 1980: David M. Burton:
*Elementary Number Theory*(revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction: Problems $1.1$: $4$