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How many perfect squares (Posted on 2006-04-25) Difficulty: 3 of 5
Suppose a and b are positive integers. We all know that a+2ab+b is a perfect square. Give an example where also a+ab+b is a perfect square. How many such examples exist?

See The Solution Submitted by Salil    
Rating: 3.0000 (2 votes)

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Solution Complete Solution | Comment 15 of 19 |

By the problem

c^2 = a^2 + ab + b^2 = (a  + b/2)^2 + 3*((b/2)^2)

where c is a positive integer

Now, substituting c= K(x^2  + 3*(y^2));  a + b/2 = K(x^2  - 3*(y^2)) and b/2 = 2Kxy and solving; we obtain b = 4Kxy giving a = K(x^2 3*(y^2) 2xy) = K(x + y) (x 3y)

So for any two given integers x and y but positive rational number K ( which is also inclusive of positive integral values) it is clearly observed that (a^2+ab+b^2) is always a perfect square whenever:

 a = K(x^2 3*(y^2) 2xy) = K(x + y) (x 3y) and

b = 4Kxy .------------------------------------------(#)

For example, K=1/4; x = 7 and y = 1 would generate the pair (a, b) = (8,7) and since 8^2 + 8*7 + 7^2 = 13^2; this is in conformity with all the conditions of the problem under reference. 

So, the given problem admits of an infinite number of solutions for the above forms corresponding to a and b in consonance with relationship (#).

Edited on April 27, 2006, 10:45 pm

Edited on April 28, 2006, 10:49 am

Edited on April 28, 2006, 10:50 am
  Posted by K Sengupta on 2006-04-27 22:44:08

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