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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.

See The Solution Submitted by K Sengupta    
Rating: 2.0000 (3 votes)

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Another solution | Comment 5 of 6 |

assume A+B=N
Since A<N, we set A+M=N, squaring gives A+2AM+M=N
thus B=2AM+M, and since B | 2A, we can set BK=2A, ie
2AMK+MK=2A.   (1)
Thus 2 | MK. Now assume that K isnt divisible by 2. Then 2 | M, and we can set M=2m, which gives.
now we see that 2 | A, and we set A=2a, which gives
but this is the same form as equation (1), thus this process will continue forever, and we have a contradiction. Thus 2 | K, and in (1) we set K=2k, which yields:

adding Mk to both sides:

thus k(k+1) must be a square, but the product of two consecutive positive integers cant be a square.

Edited on May 24, 2008, 3:53 pm
  Posted by Jonathan Lindgren on 2008-05-24 15:52:32

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