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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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Solution re(2): Adding to the solution...even numbers! | Comment 13 of 19 |
(In reply to re: Adding to the solution...even numbers! by tomarken)

Turns out that that last formula works for any multiple of 4, not just multiples of 8, but if "a" is not divisible by 8 it just produces an integer multiple of one of the odd-numbered solutions.

Here is another formula that works - again, it will produce solutions for any value of "a" that is a multiple of 4, but only those multiples of 8 will produce unique "base pairs":

b = (a^2 - 8a - 48)/16

  Posted by tomarken on 2006-04-26 12:58:02
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