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.
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2 

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5 









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3 

6 

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12 

15 









4 

8 

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5 

10 

15 

20 

25 
The easier sum is simply the left side of the formula.
Secondly, we sum along Lshapes, an example of which is shown in bold below.
1 

2 

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4 

5 









2 

4 

6 

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10 









3 

6 

9 

12 

15 









4 

8 

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16 

20 









5 

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15 

20 

25 
Note that each number in such an Lshape is a multiple of max(i,j), which ranges from 1 to N. The sum of the numbers in an Lshape 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), 118127, available here.
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