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Expression Ratio (Posted on 2013-09-03) Difficulty: 4 of 5
Determine all pairs (a,b) of positive integers with a > b, for which (a2+b2)/(a-b) is an integer which divides 1995.

Prove that there are no others.

No Solution Yet Submitted by K Sengupta    
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Solution re: Method Finished Comment 4 of 4 |
(In reply to Method by Jer)

Let n=5b.
In the quadratic formula the discriminant becomes (5b)- 4(5b)b - 4b = b
And so a = (5bb)/2 = {3b,2b}
Which explains all of the observed patterns and shows an easy way of finding a and b for any multiple of 5.

It does not prove there are no solutions where n is not a multiple of 5.

So now suppose n = 2b+c.
The discriminant becomes b+6bc+c.  The only value of c that makes this a square is 0.

  Posted by Jer on 2013-09-04 13:57:20

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