Expressing Certain Square Numbers as the Sum of Three Squares

Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house.

Robert A. HEINLEIN

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There are certain perfect square numbers that can be written as the sum of three perfect squares. In this blog post let us consider a few of these square numbers.

To begin with, consider the number 9. We can write it as

9 = 1 + 4 + 4                                        or, 3² = 1² + 2² + 2².                          [1]

This is an example of a single-digit square number (9 here). There are other square numbers also, which can be written as above. Consider the example of a two-digit square number.

49 = 4 + 9 + 36                                   or, 7² = 2² + 3² + 6²;                          [2]

Few examples from three-digit square numbers are:

169 = 9 + 16 + 144                           or, 13² = 3² + 4² + 12²;                     [3]

441 = 16 + 25 + 400                         or, 21² = 4² + 5² + 20²;                     [4]

961 = 25 + 36 + 900                         or, 31² = 5² + 6² + 30²;                     [5]

Consider few examples from four-digit perfect square numbers:

1849 = 36 + 49 + 1764                    or, 43² = 6² + 7² + 42²;

3249 = 49 + 64 + 3136                    or, 57² = 7² + 8² + 56²;

5329 = 64 + 81 + 5184                    or, 73² = 8² + 9² + 72²;

8281 = 81 + 100 + 8100                  or, 91² = 9² + 10² + 90².

From the above examples, can we see any symmetry? Consider [2], for example. The product of 2 and 3 on the right is equal to 6. In [3], the product of 3 and 4 on the right is equal to 12. Similarly, in [4] the product of 4 and 5 on the right is equal to 20. In view of this symmetry, consider the following expression:

n² + (n + 1)² + {n(n + 1)}²

= n² + (n² + 2n + 1) + {n(n + 1)}²

= 1 + 2n² + 2n + {n(n + 1)}²

= 1 + 2n(n + 1) + {n(n + 1)}²

= {1 + n(n + 1)}².

From this identity, for various positive values of n, we can obtain various relations.

Again, if we add n² in identity [2], we get the following relation:

9n² = 1n² + 4n² + 4n²                                        or, (3n)² = (1n)² + (2n)² + (2n)²                     [6]

Using this identity, we can find a few relations given below:

6² = 2² + 4² + 4²;                9² = 3² + 6² + 6²;                15² = 5² + 10² + 10² etc.

Similarly, from [3] we get the following identity:

49n² = 4n² + 9n² + 36n²                                   or, (7n)² = (2n)² + (3n)² + (6n)²;                   [7]

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