Cube as Difference between Two Squares

Theorem

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

$\displaystyle n^3$

$=$

$\displaystyle \sum_{k \mathop = 1}^n k^3 - \sum_{k \mathop = 1}^{n - 1} k^3$

$\displaystyle $

$=$

$\displaystyle \paren {\frac {n^2 \paren {n + 1}^2} 4} - \paren {\frac {\paren {n - 1}^2 n^2} 4}$

$\displaystyle $

$=$

$\displaystyle \frac {n^2 \paren {\paren {n + 1}^2 - \paren {n - 1}^2} } 4$

$\displaystyle $

$=$

$\displaystyle \paren {\frac {n \paren {n + 1} } 2}^2 - \paren {\frac {n \paren {n - 1} } 2}^2$

$\blacksquare$

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