Saturday, June 30, 2012

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.  

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