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 2² and 3² , 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 (n²-1)/2 and (n²+1)/2 e.g.  

          5² = (5²-1)/2 + (5²+1)/2.

• A triplet of three natural numbers a , b and c forms a Pythagorean Triplet , if a²+b²=c² e.g. (3,4,5) is a Pythagorean Triplet. For any natural number p greater than 1 , (2p , p²-1 , p²+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 = 2²
             3 + 6 = 9 = 3²
             6 + 10 = 16 = 4²
This shows that if we add two consecutive triangular numbers we get a square number.