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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): Partial Solution | Comment 4 of 19 |
(In reply to re: Partial Solution by Charlie)

As Charlie pointed out, the solutions can go on infinitely as long as they are integral multiples of one of the "base pairs".  Here is the formula to find the base pairs, which themselves appear to go on infinitely:

Let a = any prime number greater than or equal to 5.

b = ((a-1)/2)^2 - 1

As long as you choose a prime number for a, you can find the value for b that will make a^2 + ab + b^2 = a perfect square.

  Posted by tomarken on 2006-04-25 10:08:34
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