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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 aČ+2ab+bČ is a perfect square. Give an example where also aČ+ab+bČ is a perfect square. How many such examples exist?

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

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re(2): Complete Solution | Comment 17 of 19 |
(In reply to re: Complete Solution by Salil)

An inadvertent omission had occurred in my post giving the complete solution to the problem under reference.

While the given formula a= K(x + y)(x-3y) and b = 4Kxy does generate the desired pairs (a,b) corresponding to positive integers  x and y; the magnitude of K is POSITIVE RATIONAL NUMBER AND THEREFORE WOULD ALSO BE INCLUSIVE OF INTEGRAL VALUES. I would shortly be amending the texts constituting my original post.

Regarding your query, it would be observed that substituting

x=5; y=1 and K = 1/4 will yield the  desired pair (a,b) = (3,5) 


  Posted by K Sengupta on 2006-04-28 10:42:23
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