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Square Numbers - III

**Some interesting facts**
- There are 2n non-perfect square numbers between the squares of two consecutive natural numbers n and (n+1) e.g. between 2
^{2} and 3^{2} , there are

2 x 2 = 4 , non-perfect square numbers.
- The square number of an odd natural number n , can be expressed as the sum of two consecutive natural numbers (n
^{2}-1)/2 and (n^{2}+1)/2 e.g.

5^{2} = (5^{2}-1)/2 + (5^{2}+1)/2.
- A triplet of three natural numbers a , b and c forms a Pythagorean Triplet , if a
^{2}+b^{2}=c^{2} e.g. (3,4,5) is a Pythagorean Triplet.
For any natural number p greater than 1 , (2p , p^{2}-1 , p^{2}+1) is a Pythagorean triplet.
- The squares of numbers which have all the digits as 1 , exhibit the following pattern

- Addition of two consecutive triangular numbers exhibits the following pattern :

1 + 3 = 4 = 2^{2}
3 + 6 = 9 = 3^{2}
6 + 10 = 16 = 4^{2}
This shows that if we add two consecutive triangular numbers we get a square number.

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