Squared triangular number

For triangular numbers that are themselves square, see square triangular number.

A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes. From Gulley (2010). The nth coloured region shows n squares of dimension n by n (the rectangle is 1 evenly divided square), hence the area of the nth region is n times n × n.

In number theory, the sum of the first n cubes is the square of the nth triangular number. That is,

$1^{3}+2^{3}+3^{3}+\cdots +n^{3}=\left(1+2+3+\cdots +n\right)^{2}.$

The same equation may be written more compactly using the mathematical notation for summation:

$\sum _{k=1}^{n}k^{3}=\left(\sum _{k=1}^{n}k\right)^{2}.$

This identity is sometimes called Nicomachus's theorem, after Nicomachus of Gerasa (c. 60 – c. 120 CE).