What is the sum of 2 2

   

The sum of the first N square numbers

We consider the sum of the first N square numbers, i.e. 1 + 4 + 9 + ... + N2.
 totalFactorization 3 * total 2 * 3 * total
1     1   1  1   1  1  1   3  1  2   3
1+4     5   1  1  5  1  3  5  2  3  5
1+4+9    14    1  2  7  3  2  7  3  4  7
1+4+9+16    30    2  5  3  2  5  9etc
1+4+9+16+25    55    5  111  5  311
1+4+9+16+25+36    91    1  713  3  713
1+4+9+16+25+36+49   140    7  4  5  7  415
1+4+9+16+25+36+49+64   204    4  317  4  917
1+4+9+16+25+36+49+64+81  285  3  519   9  519
1+4+9+16+25+36+49+64+81+100  385  511  7   51121
1+4+9+16+25+36+49+64+81+100+121   50611  223 11  623
1+4+9+16+25+36+49+64+81+100+121+144   650  21325   61325
1+4+9+16+25+36+49+64+81+100+121+144+169    819 13  7  9 13  727

 

This formula is already at Brahmagupta (598-670),
in his book Brahmasphutasiddhanta (628),
but without proof.

Proofs:

  • Algebraic: With full induction
  • Geometric proof (by Giorgio Goldoni):

      Build 6 pyramids of the following shape (here for N = 4):

      They can be put together to form a cuboid with the edge lengths N, N + 1, 2N + 1.

Here is the assembly of three such pyramids:

A cuboid "with an outside staircase" is obtained.
Obviously, two such cuboids with their external stairs together form a compact cuboid!


For capital N, these pyramids resemble those pyramids known from dividing the cube into thirds by congruent pyramids:

In Chinese these are called pyramids Yang-ma, they play an important role, for example, in calculating the volume of truncated pyramids (Liu Hui, 3rd century CE, commentary on the 9 chapters).


The above pyramids, which we used in proving the formula for the sum of the first N square numbers, generalize the geometric proof for the sum of the first N numbers.

Here the case N = 5: