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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 Adding to the solution... | Comment 11 of 19 |

Expanding on yesterday's work, it turns out the the two formulas I found will generate base pairs for any odd number, not just primes.

To recap, choose any odd number, a:

b_1 = a^2 - ((a+1)/2)^2

b_2 = ((a-1)/2)^2 - 1

The pairs (a, b_1) and (a, b_2) will both produce a perfect square when a^2 + ab + b^2 is calculated.

I'll try to find a formula that will generate the base pairs where 'a' is even...

  Posted by tomarken on 2006-04-26 10:03:03
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