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Perfect Square Not (Posted on 2008-05-07) Difficulty: 2 of 5
The pairs of positive integers (A, B) are such that B divides 2A2.

Prove that A2 + B cannot be a perfect square.

  Submitted by K Sengupta    
Rating: 2.0000 (3 votes)
Solution: (Hide)
At the outset, we substitute 2A2 = c*B. Since B divides 2A2, it follows that c must be a positive integer.

If possible, let us suppose that:

A2 + B = G2, for some positive integer G

Then, we have:

A2*c2 + B*c2 = G2*c2

or, A2*c2 + 2A2*c = G2*c2

or, A2(c2 + 2c) = (G*c)2......(*)

But, c2 < c2 + 2c < (c+1)2, and accordingly, the lhs of (*) cannot be the square of a positive integer. This leads to a contradiction.

Consequently, A2 + B cannot be a perfect square.

***************************************

For an alternative methodology, refer to the solution submitted by Praneeth in this location.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re: SolutionK Sengupta2008-05-26 14:13:59
Another solutionJonathan Lindgren2008-05-24 15:52:32
SolutionSolutionPraneeth2008-05-09 04:00:07
Some Thoughtsre: To B not a square >>>. RevisitedAdy TZIDON2008-05-07 19:15:48
To B not a squareGamer2008-05-07 17:22:09
SolutionNO WAYAdy TZIDON2008-05-07 11:14:10
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