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Simultaneous Surmise (Posted on 2014-11-29) Difficulty: 4 of 5
Each of X and Y is a positive integer.

Can X2 +Y+2 and Y2 + 4X be simultaneously perfect squares?

If so, provide an example.
If not, prove it.

No Solution Yet Submitted by K Sengupta    
Rating: 3.6667 (3 votes)

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Solution Finally! | Comment 1 of 3
There is a puzzle today! I will solve it.

The first equation implies Y+2>=2X+1, and the second equation implies 4X>=2Y+1. Therefore, 2Y+4>=4X+2>=2Y+3. Since 4X+2 is even, 2Y+4=4X+2. That means that Y=2X-1. However, Y^2+4X=Y^2+2Y+2, which is never a square for positive Y. Therefore, it is impossible to have X^2+Y+2 and Y^2+4X both be squares.

  Posted by Math Man on 2014-11-29 10:16:30
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