This is the most elementary of my favourite formulas. I think I first saw this as a Year 8 student and have been enthralled by it ever since.

In words, the sum of the first N cubes is the square of the sum of the first N squares. Furthermore, both sides are equal to the square of the binomial coefficient .

The formula was discovered by Aryabhata of Patna around 1500 years ago, but it may have been known before then. The most elementary proof is probably one by mathematical induction:

Let S be the set of positive integers N for which . We wish to show that S is the set of all positive integers . Firstly since for N=1, . Assume . That is,

For convenience denote by the sum . Then

This shows that if , then . Hence by the principle of mathematical induction, and we are done.

Next I will show the nicest proof that I am aware of (see p126 of [1] which also contains a proof by picture). We form the grid of numbers of the form for i and j from 1 to N (as you would see in the multiplication tables) and sum the numbers of the grid in two ways. Below is an example for the case N=5.

1 | 2 | 3 | 4 | 5 | ||||

2 | 4 | 6 | 8 | 10 | ||||

3 | 6 | 9 | 12 | 15 | ||||

4 | 8 | 12 | 16 | 20 | ||||

5 | 10 | 15 | 20 | 25 |

The easier sum is simply the left side of the formula.

Secondly, we sum along L-shapes, an example of which is shown in bold below.

1 | 2 | 3 | 4 |
5 | ||||

2 | 4 | 6 | 8 |
10 | ||||

3 | 6 | 9 | 12 |
15 | ||||

4 |
8 |
12 |
16 |
20 | ||||

5 | 10 | 15 | 20 | 25 |

Note that each number in such an L-shape is a multiple of max(i,j), which ranges from 1 to N. The sum of the numbers in an L-shape is then this multiple times the sum of 1, 2, …, max(i,j)-1, max(i,j), max(i,j) – 1, …, 2, 1, which can easily be shown to be (think of counting the points of a square grid along diagonals). Hence the total is

.

In other words,

Reference:

[1] C. Alsina and R. Nelsen, “An Invitation to Proofs Without Words”, Eur. J. Pure Appl. Math, 3 (2010), 118-127, available here.

## Leave a Reply