*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**

Welcome to the blog **Math1089 – Mathematics for All**.

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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, 3^{2} = 1^{2} + 2^{2} + 2^{2}. [**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, 7^{2} = 2^{2} + 3^{2} + 6^{2}; [**2**]

Few examples from three-digit square numbers are:

169 = 9 + 16 + 144 or, 13^{2} = 3^{2} + 4^{2} + 12^{2}; [**3**]

441 = 16 + 25 + 400 or, 21^{2} = 4^{2} + 5^{2} + 20^{2}; [**4**]

961 = 25 + 36 + 900 or, 31^{2} = 5^{2} + 6^{2} + 30^{2}; [**5**]

Consider few examples from four-digit perfect square numbers:

1849 = 36 + 49 + 1764 or, 43^{2} = 6^{2} + 7^{2} + 42^{2};

3249 = 49 + 64 + 3136 or, 57^{2} = 7^{2} + 8^{2} + 56^{2};

5329 = 64 + 81 + 5184 or, 73^{2} = 8^{2} + 9^{2} + 72^{2};

8281 = 81 + 100 + 8100 or, 91^{2} = 9^{2} + 10^{2} + 90^{2}.

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:

n^{2}+ (n+ 1)^{2}+ {n(n+ 1)}^{2}=

n^{2}+ (n^{2}+ 2n+ 1) + {n(n+ 1)}^{2}= 1 + 2

n^{2}+ 2n+ {n(n+ 1)}^{2}= 1 + 2

n(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 *n*^{2} in identity [**2**], we get the following relation:

9*n*^{2} = 1*n*^{2} + 4*n*^{2} + 4*n*^{2} or, (3*n*)^{2} = (1*n*)^{2} + (2*n*)^{2} + (2*n*)^{2} [**6**]

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

6^{2} = 2^{2} + 4^{2} + 4^{2}; 9^{2} = 3^{2} + 6^{2} + 6^{2}; 15^{2} = 5^{2} + 10^{2} + 10^{2} etc.

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

49*n*^{2} = 4*n*^{2} + 9*n*^{2} + 36*n*^{2} or, (7*n*)^{2} = (2*n*)^{2} + (3*n*)^{2} + (6*n*)^{2}; [**7**]

Your suggestions are eagerly and respectfully welcome! See you soon with a new mathematics blog that you and I call **“****Math1089 – Mathematics for All!**“.